# category_property_assignments

category_id: Alg(R)
property_id: strict initial object
reason:
Assume for contradiction that it has a strict initial object. Then it would have disjoint finite products and would have a strict initial object, so it would be thin (by <a href="/category-implication/dual_disjoint_coproducts_strict">this result</a>) – contradiction.


category_id: Alg(R)
property_id: coextensive
reason:
Assume for contradiction that it is coextensive. Then it would be coextensive and would have finite coproducts, so it would be codistributive (by <a href="/category-implication/dual_extensive_distributivity">this result</a>) – contradiction.


category_id: Alg(R)
property_id: CSP
reason:
Assume for contradiction that it satisfies CSP. Then it would have zero morphisms (by <a href="/category-implication/dual_CIP_assumption">this result</a>) – contradiction.


category_id: Alg(R)
property_id: cogenerator
reason:
Assume for contradiction that it has a cogenerator. Then it would have a cogenerating set (by <a href="/category-implication/dual_generator_consequence">this result</a>) – contradiction.


category_id: Alg(R)
property_id: epi-regular
reason:
Assume for contradiction that it is epi-regular. Then it would be balanced (by <a href="/category-implication/dual_mono_regular_consequence">this result</a>) – contradiction.


category_id: Alg(R)
property_id: conormal
reason:
Assume for contradiction that it is conormal. Then it would have zero morphisms (by <a href="/category-implication/dual_normal_condition">this result</a>) – contradiction.


category_id: Alg(R)
property_id: coaccessible
reason:
Assume for contradiction that it is coaccessible. Then it would have a cogenerating set (by <a href="/category-implication/dual_accessible_trivial_consequence">this result</a>) – contradiction.


category_id: Alg(R)
property_id: quotient object classifier
reason:
Assume for contradiction that it has a quotient object classifier. Then it would be epi-regular (by <a href="/category-implication/dual_subobject_classifier_consequence">this result</a>) – contradiction.


category_id: Alg(R)
property_id: counital
reason:
Assume for contradiction that it is counital. Then it would be pointed (by <a href="/category-implication/dual_unital_assumptions">this result</a>) – contradiction.


category_id: Alg(R)
property_id: self-dual
reason:
Assume for contradiction that it is self-dual. Then it would be self-dual and would have a strict terminal object, so it would have a strict initial object (by <a href="/category-implication/self-dual_strict terminal object">this result</a>) – contradiction.


category_id: Ban
property_id: zero morphisms
reason:
Since it is pointed, it has zero morphisms (by <a href="/category-implication/pointed_characterization">this result</a>).


category_id: Ban
property_id: filtered colimits
reason:
Since it has cartesian filtered colimits, it has filtered colimits (by <a href="/category-implication/cartesian_filtered_colimits_condition">this result</a>).


category_id: Ban
property_id: finite products
reason:
Since it has cartesian filtered colimits, it has finite products (by <a href="/category-implication/cartesian_filtered_colimits_condition">this result</a>).


category_id: Ban
property_id: products
reason:
Since it satisfies CIP, it has products (by <a href="/category-implication/CIP_assumption">this result</a>).


category_id: Ban
property_id: cocomplete
reason:
Since it is locally ℵ₁-presentable, it is cocomplete (by <a href="/category-implication/locally_presentable_definition_countable">this result</a>).


category_id: Ban
property_id: generating set
reason:
Since it is accessible, it has a generating set (by <a href="/category-implication/accessible_trivial_consequence">this result</a>).


category_id: Ban
property_id: generator
reason:
Since it has coproducts and has a generating set and has zero morphisms, it has a generator (by <a href="/category-implication/generator_via_coproduct">this result</a>).


category_id: Ban
property_id: Cauchy complete
reason:
Since it is accessible, it is Cauchy complete (by <a href="/category-implication/accessible_require_Cauchy_complete">this result</a>).


category_id: Ban
property_id: cokernels
reason:
Since it has coequalizers and has zero morphisms, it has cokernels (by <a href="/category-implication/dual_kernels_criterion">this result</a>).


category_id: Ban
property_id: equalizers
reason:
Since it has connected limits, it has equalizers (by <a href="/category-implication/connected_limits_characterization">this result</a>).


category_id: Ban
property_id: complete
reason:
Since it has equalizers and has products, it is complete (by <a href="/category-implication/complete_characterization">this result</a>).


category_id: Ban
property_id: finitely complete
reason:
Since it is complete, it is finitely complete (by <a href="/category-implication/complete_consequence">this result</a>).


category_id: Ban
property_id: kernels
reason:
Since it has equalizers and has zero morphisms, it has kernels (by <a href="/category-implication/kernels_criterion">this result</a>).


category_id: Ban
property_id: biproducts
reason:
Assume for contradiction that it has biproducts. Then it would have biproducts and would be finitely complete, so it would be unital (by <a href="/category-implication/biproducts_unital">this result</a>) – contradiction.


category_id: Ban
property_id: exact filtered colimits
reason:
Assume for contradiction that it has exact filtered colimits. Then it would have filtered-colimit-stable monomorphisms (by <a href="/category-implication/exact_filtered_colimits_monos">this result</a>) – contradiction.


category_id: Ban
property_id: left cancellative
reason:
Assume for contradiction that it is left cancellative. Then it would have filtered colimits and would be left cancellative, so it would have filtered-colimit-stable monomorphisms (by <a href="/category-implication/filtered_monos_trivial">this result</a>) – contradiction.


category_id: Ban
property_id: additive
reason:
Assume for contradiction that it is additive. Then it would have biproducts (by <a href="/category-implication/additive_biproducts">this result</a>) – contradiction.


category_id: Ban
property_id: preadditive
reason:
Assume for contradiction that it is preadditive. Then it would have finite products and would be preadditive, so it would be additive (by <a href="/category-implication/additive_definition">this result</a>) – contradiction.


category_id: Ban
property_id: abelian
reason:
Assume for contradiction that it is abelian. Then it would be additive (by <a href="/category-implication/abelian_definition">this result</a>) – contradiction.


category_id: Ban
property_id: locally finitely presentable
reason:
Assume for contradiction that it is locally finitely presentable. Then it would have exact filtered colimits (by <a href="/category-implication/locally_finitely_presentable_consequence">this result</a>) – contradiction.


category_id: Ban
property_id: locally strongly finitely presentable
reason:
Assume for contradiction that it is locally strongly finitely presentable. Then it would be locally finitely presentable (by <a href="/category-implication/locally_strongly_finitely_presentable_raise">this result</a>) – contradiction.


category_id: Ban
property_id: finitary algebraic
reason:
Assume for contradiction that it is finitary algebraic. Then it would be locally strongly finitely presentable (by <a href="/category-implication/finitary_algebraic_consequence">this result</a>) – contradiction.


category_id: Ban
property_id: finitely accessible
reason:
Assume for contradiction that it is finitely accessible. Then it would be cocomplete and would be finitely accessible, so it would be locally finitely presentable (by <a href="/category-implication/locally_presentable_definition_finite">this result</a>) – contradiction.


category_id: Ban
property_id: generalized variety
reason:
Assume for contradiction that it is a generalized variety. Then it would be cocomplete and would be a generalized variety, so it would be locally strongly finitely presentable (by <a href="/category-implication/locally_strongly_finitely_presentable_definition">this result</a>) – contradiction.


category_id: Ban
property_id: locally finitely multi-presentable
reason:
Assume for contradiction that it is locally finitely multi-presentable. Then it would be finitely accessible (by <a href="/category-implication/locally_finitely_multi-presentable_definition">this result</a>) – contradiction.


category_id: Ban
property_id: multi-algebraic
reason:
Assume for contradiction that it is multi-algebraic. Then it would be locally finitely multi-presentable (by <a href="/category-implication/multi-algebraic_implies_locally_finitely_multi-presentable">this result</a>) – contradiction.


category_id: Ban
property_id: Malcev
reason:
Assume for contradiction that it is Malcev. Then it would be Malcev and would be pointed, so it would be unital (by <a href="/category-implication/malcev_implies_unital">this result</a>) – contradiction.


category_id: Ban
property_id: conormal
reason:
Assume for contradiction that it is conormal. Then it would be epi-regular (by <a href="/category-implication/dual_mono_regular_via_kernels">this result</a>) – contradiction.


category_id: Ban
property_id: self-dual
reason:
Assume for contradiction that it is self-dual. Then it would satisfy CIP and would be self-dual, so it would satisfy CSP (by <a href="/category-implication/self-dual_CIP">this result</a>) – contradiction.


category_id: Sp
property_id: generating set
reason:
Since it is essentially small, it has a generating set (by <a href="/category-implication/essentially_small_consequences">this result</a>).


category_id: Sp
property_id: locally essentially small
reason:
Since it is essentially small, it is locally essentially small (by <a href="/category-implication/essentially_small_consequences">this result</a>).


category_id: Sp
property_id: well-copowered
reason:
Since it is essentially small, it is well-copowered (by <a href="/category-implication/essentially_small_consequences">this result</a>).


category_id: Sp
property_id: well-powered
reason:
Since it is essentially small, it is well-powered (by <a href="/category-implication/essentially_small_consequences">this result</a>).


category_id: Sp
property_id: cartesian closed
reason:
Since it is an elementary topos, it is cartesian closed (by <a href="/category-implication/topos_definition">this result</a>).


category_id: Sp
property_id: finite products
reason:
Since it is cartesian closed, it has finite products (by <a href="/category-implication/ccc_condition">this result</a>).


category_id: Sp
property_id: finitely complete
reason:
Since it is an elementary topos, it is finitely complete (by <a href="/category-implication/topos_definition">this result</a>).


category_id: Sp
property_id: accessible
reason:
Since it is Cauchy complete and is essentially small, it is accessible (by <a href="/category-implication/small_accessible_characterization">this result</a>).


category_id: Sp
property_id: subobject classifier
reason:
Since it is an elementary topos, it has a subobject classifier (by <a href="/category-implication/topos_definition">this result</a>).


category_id: Sp
property_id: mono-regular
reason:
Since it has a subobject classifier, it is mono-regular (by <a href="/category-implication/subobject_classifier_consequence">this result</a>).


category_id: Sp
property_id: regular subobject classifier
reason:
Since it has a subobject classifier, it has a regular subobject classifier (by <a href="/category-implication/regular_subobject_weaker">this result</a>).


category_id: Sp
property_id: disjoint finite coproducts
reason:
Since it is an elementary topos, it has disjoint finite coproducts (by <a href="/category-implication/topos_consequence">this result</a>).


category_id: Sp
property_id: finite coproducts
reason:
Since it has disjoint finite coproducts, it has finite coproducts (by <a href="/category-implication/disjoint_coproducts_consequence">this result</a>).


category_id: Sp
property_id: distributive
reason:
Since it is cartesian closed and has finite coproducts, it is distributive (by <a href="/category-implication/distributive_criterion">this result</a>).


category_id: Sp
property_id: strict initial object
reason:
Since it is distributive, it has a strict initial object (by <a href="/category-implication/distributive_consequence">this result</a>).


category_id: Sp
property_id: initial object
reason:
Since it has a strict initial object, it has an initial object (by <a href="/category-implication/strict_initial_consequence">this result</a>).


category_id: Sp
property_id: epi-regular
reason:
Since it is an elementary topos, it is epi-regular (by <a href="/category-implication/topos_consequence">this result</a>).


category_id: Sp
property_id: finitely cocomplete
reason:
Since it is an elementary topos, it is finitely cocomplete (by <a href="/category-implication/topos_consequence">this result</a>).


category_id: Sp
property_id: locally cartesian closed
reason:
Since it is an elementary topos, it is locally cartesian closed (by <a href="/category-implication/topos_is_locally_cartesian_closed">this result</a>).


category_id: Sp
property_id: coregular
reason:
Since it is an elementary topos, it is coregular (by <a href="/category-implication/topos_implies_coregular">this result</a>).


category_id: Sp
property_id: extensive
reason:
Since it has disjoint finite coproducts and is locally cartesian closed, it is extensive (by <a href="/category-implication/lcc_implies_extensive">this result</a>).


category_id: Sp
property_id: co-Malcev
reason:
Since it is an elementary topos, it is co-Malcev (by <a href="/category-implication/topos_is_malcev">this result</a>).


category_id: Sp
property_id: inhabited
reason:
Since it is connected, it is inhabited (by <a href="/category-implication/connected_consequence">this result</a>).


category_id: Sp
property_id: coequalizers
reason:
Since it is finitely cocomplete, it has coequalizers (by <a href="/category-implication/dual_finitely_complete_characterization">this result</a>).


category_id: Sp
property_id: regular
reason:
Since it has coequalizers and is finitely complete and is locally cartesian closed, it is regular (by <a href="/category-implication/lcc_implies_regular">this result</a>).


category_id: Sp
property_id: reflexive coequalizers
reason:
Since it has coequalizers, it has reflexive coequalizers (by <a href="/category-implication/dual_coreflexive_equalizers_are_equalizers">this result</a>).


category_id: Sp
property_id: multi-initial object
reason:
Since it has an initial object, it has a multi-initial object (by <a href="/category-implication/dual_multi-terminal_with_connected">this result</a>).


category_id: Sp
property_id: cogenerating set
reason:
Since it is essentially small, it has a cogenerating set (by <a href="/category-implication/dual_essentially_small_consequences">this result</a>).


category_id: Sp
property_id: coaccessible
reason:
Since it is Cauchy complete and is essentially small, it is coaccessible (by <a href="/category-implication/dual_small_accessible_characterization">this result</a>).


category_id: Sp
property_id: countable products
reason:
Assume for contradiction that it has countable products. Then it would have countable powers (by <a href="/category-implication/countable_products_consequence">this result</a>) – contradiction.


category_id: Sp
property_id: products
reason:
Assume for contradiction that it has products. Then it would have countable products (by <a href="/category-implication/products_consequences">this result</a>) – contradiction.


category_id: Sp
property_id: cofiltered limits
reason:
Assume for contradiction that it has cofiltered limits. Then it would have cofiltered limits and would have finite products, so it would have products (by <a href="/category-implication/products_criterion">this result</a>) – contradiction.


category_id: Sp
property_id: complete
reason:
Assume for contradiction that it is complete. Then it would have products (by <a href="/category-implication/complete_characterization">this result</a>) – contradiction.


category_id: Sp
property_id: wide pullbacks
reason:
Assume for contradiction that it has wide pullbacks. Then it would have a terminal object and would have wide pullbacks, so it would be complete (by <a href="/category-implication/complete_criterion">this result</a>) – contradiction.


category_id: Sp
property_id: connected limits
reason:
Assume for contradiction that it has connected limits. Then it would have wide pullbacks (by <a href="/category-implication/connected_limits_characterization">this result</a>) – contradiction.


category_id: Sp
property_id: essentially finite
reason:
Assume for contradiction that it is essentially finite. Then it would be essentially finite and would have pullbacks, so it would have wide pullbacks (by <a href="/category-implication/wide_pullbacks_finite">this result</a>) – contradiction.


category_id: Sp
property_id: sequential limits
reason:
Assume for contradiction that it has sequential limits. Then it would have finite products and would have sequential limits, so it would have countable products (by <a href="/category-implication/countable_products_criterion">this result</a>) – contradiction.


category_id: Sp
property_id: directed limits
reason:
Assume for contradiction that it has directed limits. Then it would have sequential limits (by <a href="/category-implication/directed_limits_consequence">this result</a>) – contradiction.


category_id: Sp
property_id: powers
reason:
Assume for contradiction that it has powers. Then it would have countable powers (by <a href="/category-implication/powers_include_countable_powers">this result</a>) – contradiction.


category_id: Sp
property_id: multi-complete
reason:
Assume for contradiction that it is multi-complete. Then it would have finite coproducts and would be multi-complete, so it would be complete (by <a href="/category-implication/multi-complete_with_finite_coproducts">this result</a>) – contradiction.


category_id: Sp
property_id: copowers
reason:
Assume for contradiction that it has copowers. Then it would be cartesian closed and would have copowers, so it would have powers (by <a href="/category-implication/power_construction">this result</a>) – contradiction.


category_id: Sp
property_id: CIP
reason:
Assume for contradiction that it satisfies CIP. Then it would have products (by <a href="/category-implication/CIP_assumption">this result</a>) – contradiction.


category_id: Sp
property_id: small
reason:
Assume for contradiction that it is small. Then it would be locally small (by <a href="/category-implication/small_consequence">this result</a>) – contradiction.


category_id: Sp
property_id: finitary algebraic
reason:
Assume for contradiction that it is finitary algebraic. Then it would have a generator (by <a href="/category-implication/free-algebra-generates">this result</a>) – contradiction.


category_id: Sp
property_id: groupoid
reason:
Assume for contradiction that it is a groupoid. Then it would have directed limits (by <a href="/category-implication/groupoid_consequence">this result</a>) – contradiction.


category_id: Sp
property_id: Grothendieck abelian
reason:
Assume for contradiction that it is Grothendieck abelian. Then it would have a generator (by <a href="/category-implication/grothendieck_abelian_definition">this result</a>) – contradiction.


category_id: Sp
property_id: pointed
reason:
Assume for contradiction that it is pointed. Then it would be pointed and would have a strict initial object, so it would be trivial (by <a href="/category-implication/strict_initial_trivial">this result</a>) – contradiction.


category_id: Sp
property_id: zero morphisms
reason:
Assume for contradiction that it has zero morphisms. Then it would have an initial object and would have zero morphisms, so it would be pointed (by <a href="/category-implication/pointed_characterization">this result</a>) – contradiction.


category_id: Sp
property_id: normal
reason:
Assume for contradiction that it is normal. Then it would have zero morphisms (by <a href="/category-implication/normal_condition">this result</a>) – contradiction.


category_id: Sp
property_id: preadditive
reason:
Assume for contradiction that it is preadditive. Then it would be mono-regular and would be preadditive, so it would be normal (by <a href="/category-implication/normal_criterion">this result</a>) – contradiction.


category_id: Sp
property_id: locally multi-presentable
reason:
Assume for contradiction that it is locally multi-presentable. Then it would have connected limits (by <a href="/category-implication/locally_multi-presentable_definition">this result</a>) – contradiction.


category_id: Sp
property_id: multi-cocomplete
reason:
Assume for contradiction that it is multi-cocomplete. Then it would be accessible and would be multi-cocomplete, so it would be locally multi-presentable (by <a href="/category-implication/locally_multi-presentable_another_definition">this result</a>) – contradiction.


category_id: Sp
property_id: locally finitely multi-presentable
reason:
Assume for contradiction that it is locally finitely multi-presentable. Then it would have connected limits (by <a href="/category-implication/locally_finitely_multi-presentable_definition">this result</a>) – contradiction.


category_id: Sp
property_id: locally poly-presentable
reason:
Assume for contradiction that it is locally poly-presentable. Then it would have wide pullbacks (by <a href="/category-implication/locally_poly-presentable_definition">this result</a>) – contradiction.


category_id: Sp
property_id: locally strongly finitely presentable
reason:
Assume for contradiction that it is locally strongly finitely presentable. Then it would be multi-algebraic (by <a href="/category-implication/varieties_are_multi-algebraic">this result</a>) – contradiction.


category_id: Sp
property_id: countable coproducts
reason:
Assume for contradiction that it has countable coproducts. Then it would be cartesian closed and would have countable coproducts, so it would be countably distributive (by <a href="/category-implication/countably_distributive_criterion">this result</a>) – contradiction.


category_id: Sp
property_id: coproducts
reason:
Assume for contradiction that it has coproducts. Then it would be cartesian closed and would have coproducts, so it would be infinitary distributive (by <a href="/category-implication/infinitary_distributive_criterion">this result</a>) – contradiction.


category_id: Sp
property_id: disjoint coproducts
reason:
Assume for contradiction that it has disjoint coproducts. Then it would have coproducts (by <a href="/category-implication/disjoint_coproducts_equivalence">this result</a>) – contradiction.


category_id: Sp
property_id: infinitary extensive
reason:
Assume for contradiction that it is infinitary extensive. Then it would have coproducts (by <a href="/category-implication/infinitary_extensive_assumption">this result</a>) – contradiction.


category_id: Sp
property_id: Grothendieck topos
reason:
Assume for contradiction that it is a Grothendieck topos. Then it would have coproducts (by <a href="/category-implication/grothendieck_topos_definition">this result</a>) – contradiction.


category_id: Sp
property_id: cocomplete
reason:
Assume for contradiction that it is cocomplete. Then it would be cocomplete and would be extensive and would be locally cartesian closed, so it would be infinitary extensive (by <a href="/category-implication/lcc_extensive_yields_infinitary">this result</a>) – contradiction.


category_id: Sp
property_id: locally finitely presentable
reason:
Assume for contradiction that it is locally finitely presentable. Then it would be cocomplete (by <a href="/category-implication/locally_presentable_definition_finite">this result</a>) – contradiction.


category_id: Sp
property_id: locally ℵ₁-presentable
reason:
Assume for contradiction that it is locally ℵ₁-presentable. Then it would be cocomplete (by <a href="/category-implication/locally_presentable_definition_countable">this result</a>) – contradiction.


category_id: Sp
property_id: locally presentable
reason:
Assume for contradiction that it is locally presentable. Then it would be cocomplete (by <a href="/category-implication/locally_presentable_definition">this result</a>) – contradiction.


category_id: Sp
property_id: cosifted limits
reason:
Assume for contradiction that it has cosifted limits. Then it would have cofiltered limits (by <a href="/category-implication/dual_sifted_colimits_consequence">this result</a>) – contradiction.


category_id: Sp
property_id: filtered colimits
reason:
Assume for contradiction that it has filtered colimits. Then it would have filtered colimits and would have finite coproducts, so it would have coproducts (by <a href="/category-implication/dual_products_criterion">this result</a>) – contradiction.


category_id: Sp
property_id: sifted colimits
reason:
Assume for contradiction that it has sifted colimits. Then it would have filtered colimits (by <a href="/category-implication/sifted_colimits_consequence">this result</a>) – contradiction.


category_id: Sp
property_id: connected colimits
reason:
Assume for contradiction that it has connected colimits. Then it would have sifted colimits (by <a href="/category-implication/sifted_categories_are_connected">this result</a>) – contradiction.


category_id: Sp
property_id: directed colimits
reason:
Assume for contradiction that it has directed colimits. Then it would have filtered colimits (by <a href="/category-implication/directed_colimits_suffice">this result</a>) – contradiction.


category_id: Sp
property_id: exact filtered colimits
reason:
Assume for contradiction that it has exact filtered colimits. Then it would have filtered colimits (by <a href="/category-implication/exact_filtered_colimits_condition">this result</a>) – contradiction.


category_id: Sp
property_id: cartesian filtered colimits
reason:
Assume for contradiction that it has cartesian filtered colimits. Then it would have filtered colimits (by <a href="/category-implication/cartesian_filtered_colimits_condition">this result</a>) – contradiction.


category_id: Sp
property_id: filtered-colimit-stable monomorphisms
reason:
Assume for contradiction that it has filtered-colimit-stable monomorphisms. Then it would have filtered colimits (by <a href="/category-implication/filtered_monos_assumption">this result</a>) – contradiction.


category_id: Sp
property_id: finitely accessible
reason:
Assume for contradiction that it is finitely accessible. Then it would have filtered colimits (by <a href="/category-implication/accessible_require_filtered_colimit">this result</a>) – contradiction.


category_id: Sp
property_id: generalized variety
reason:
Assume for contradiction that it is a generalized variety. Then it would have sifted colimits (by <a href="/category-implication/generalized_variety_require_sifted_colimit">this result</a>) – contradiction.


category_id: Sp
property_id: wide pushouts
reason:
Assume for contradiction that it has wide pushouts. Then it would have coequalizers and would have wide pushouts, so it would have connected colimits (by <a href="/category-implication/dual_connected_limits_characterization">this result</a>) – contradiction.


category_id: Sp
property_id: sequential colimits
reason:
Assume for contradiction that it has sequential colimits. Then it would have finite coproducts and would have sequential colimits, so it would have countable coproducts (by <a href="/category-implication/dual_countable_products_criterion">this result</a>) – contradiction.


category_id: Sp
property_id: disjoint products
reason:
Assume for contradiction that it has disjoint products. Then it would have products (by <a href="/category-implication/dual_disjoint_coproducts_equivalence">this result</a>) – contradiction.


category_id: Sp
property_id: disjoint finite products
reason:
Assume for contradiction that it has disjoint finite products. Then it would have disjoint finite products and would have a strict initial object, so it would be thin (by <a href="/category-implication/dual_disjoint_coproducts_strict">this result</a>) – contradiction.


category_id: Sp
property_id: exact cofiltered limits
reason:
Assume for contradiction that it has exact cofiltered limits. Then it would have cofiltered limits (by <a href="/category-implication/dual_exact_filtered_colimits_condition">this result</a>) – contradiction.


category_id: Sp
property_id: cocartesian cofiltered limits
reason:
Assume for contradiction that it has cocartesian cofiltered limits. Then it would have cofiltered limits (by <a href="/category-implication/dual_cartesian_filtered_colimits_condition">this result</a>) – contradiction.


category_id: Sp
property_id: infinitary codistributive
reason:
Assume for contradiction that it is infinitary codistributive. Then it would have products (by <a href="/category-implication/dual_infinitary_distributive_assumption">this result</a>) – contradiction.


category_id: Sp
property_id: countably codistributive
reason:
Assume for contradiction that it is countably codistributive. Then it would have countable products (by <a href="/category-implication/dual_countably_distributive_assumption">this result</a>) – contradiction.


category_id: Sp
property_id: infinitary coextensive
reason:
Assume for contradiction that it is infinitary coextensive. Then it would have products (by <a href="/category-implication/dual_infinitary_extensive_assumption">this result</a>) – contradiction.


category_id: Sp
property_id: coextensive
reason:
Assume for contradiction that it is coextensive. Then it would have disjoint finite products (by <a href="/category-implication/dual_extensive_consequences">this result</a>) – contradiction.


category_id: Sp
property_id: CSP
reason:
Assume for contradiction that it satisfies CSP. Then it would have coproducts (by <a href="/category-implication/dual_CIP_assumption">this result</a>) – contradiction.


category_id: Sp
property_id: cofiltered-limit-stable epimorphisms
reason:
Assume for contradiction that it has cofiltered-limit-stable epimorphisms. Then it would have cofiltered limits (by <a href="/category-implication/dual_filtered_monos_assumption">this result</a>) – contradiction.


category_id: Sp
property_id: locally copresentable
reason:
Assume for contradiction that it is locally copresentable. Then it would be complete (by <a href="/category-implication/dual_locally_presentable_definition">this result</a>) – contradiction.


category_id: Sp
property_id: regular quotient object classifier
reason:
Assume for contradiction that it has a regular quotient object classifier. Then it would have a regular quotient object classifier and would have a strict initial object, so it would be thin (by <a href="/category-implication/dual_regular_subobject_classifier_strict">this result</a>) – contradiction.


category_id: Sp
property_id: self-dual
reason:
Assume for contradiction that it is self-dual. Then it would be self-dual and would have a strict initial object, so it would have a strict terminal object (by <a href="/category-implication/self-dual_strict initial object">this result</a>) – contradiction.


category_id: CAlg(R)
property_id: disjoint finite products
reason:
Since it is coextensive, it has disjoint finite products (by <a href="/category-implication/dual_extensive_consequences">this result</a>).


category_id: CAlg(R)
property_id: codistributive
reason:
Since it is coextensive and has finite coproducts, it is codistributive (by <a href="/category-implication/dual_extensive_distributivity">this result</a>).


category_id: CAlg(R)
property_id: extensive
reason:
Assume for contradiction that it is extensive. Then it would have disjoint finite coproducts (by <a href="/category-implication/extensive_consequences">this result</a>) – contradiction.


category_id: CAlg(R)
property_id: cocartesian coclosed
reason:
Assume for contradiction that it is cocartesian coclosed. Then it would be cocartesian coclosed and would have countable products, so it would be countably codistributive (by <a href="/category-implication/dual_countably_distributive_criterion">this result</a>) – contradiction.


category_id: CAlg(R)
property_id: cocartesian cofiltered limits
reason:
Assume for contradiction that it has cocartesian cofiltered limits. Then it would have cocartesian cofiltered limits and would be codistributive and would have products, so it would be infinitary codistributive (by <a href="/category-implication/dual_infinite_distributive_filtered_criterion">this result</a>) – contradiction.


category_id: CAlg(R)
property_id: infinitary coextensive
reason:
Assume for contradiction that it is infinitary coextensive. Then it would have finite coproducts and would be infinitary coextensive, so it would be infinitary codistributive (by <a href="/category-implication/dual_infinitary_extensive_distributivity">this result</a>) – contradiction.


category_id: CMon
property_id: finite products
reason:
Since it has biproducts, it has finite products (by <a href="/category-implication/biproducts_definition_consequences">this result</a>).


category_id: CMon
property_id: zero morphisms
reason:
Since it has biproducts, it has zero morphisms (by <a href="/category-implication/biproducts_definition_consequences">this result</a>).


category_id: CMon
property_id: cogenerating set
reason:
Assume for contradiction that it has a cogenerating set. Then it would have a cogenerating set and would have products and would have zero morphisms, so it would have a cogenerator (by <a href="/category-implication/dual_generator_via_coproduct">this result</a>) – contradiction.


category_id: CMon
property_id: self-dual
reason:
Assume for contradiction that it is self-dual. Then it would have exact filtered colimits and would be self-dual, so it would have exact cofiltered limits (by <a href="/category-implication/self-dual_exact filtered colimits">this result</a>) – contradiction.


category_id: Set_c
property_id: countable coproducts
reason:
Since it is countably distributive, it has countable coproducts (by <a href="/category-implication/countably_distributive_assumption">this result</a>).


category_id: Set_c
property_id: finite products
reason:
Since it is countably distributive, it has finite products (by <a href="/category-implication/countably_distributive_assumption">this result</a>).


category_id: Set_c
property_id: distributive
reason:
Since it is countably distributive, it is distributive (by <a href="/category-implication/countably_distributive_trivial">this result</a>).


category_id: Set_c
property_id: finite coproducts
reason:
Since it is distributive, it has finite coproducts (by <a href="/category-implication/distributive_assumption">this result</a>).


category_id: Set_c
property_id: disjoint finite coproducts
reason:
Since it is extensive, it has disjoint finite coproducts (by <a href="/category-implication/extensive_consequences">this result</a>).


category_id: Set_c
property_id: natural numbers object
reason:
Since it is countably distributive, it has a natural numbers object (by <a href="/category-implication/nno_criterion">this result</a>).


category_id: Set_c
property_id: finitely cocomplete
reason:
Since it is coregular, it is finitely cocomplete (by <a href="/category-implication/dual_regular_def">this result</a>).


category_id: Set_c
property_id: sequential colimits
reason:
Since it has coequalizers and has countable coproducts, it has sequential colimits (by <a href="/category-implication/dual_sequential_limits_criterion">this result</a>).


category_id: Set_c
property_id: cartesian closed
reason:
Assume for contradiction that it is cartesian closed. Then it would be cartesian closed and would have countable copowers, so it would have countable powers (by <a href="/category-implication/countable_power_construction">this result</a>) – contradiction.


category_id: Set_c
property_id: locally ℵ₁-presentable
reason:
Assume for contradiction that it is locally ℵ₁-presentable. Then it would be ℵ₁-accessible (by <a href="/category-implication/locally_presentable_definition_countable">this result</a>) – contradiction.


category_id: Set_c
property_id: locally finitely presentable
reason:
Assume for contradiction that it is locally finitely presentable. Then it would be locally ℵ₁-presentable (by <a href="/category-implication/locally_finitely_presentable_raise">this result</a>) – contradiction.


category_id: Set_c
property_id: finitely accessible
reason:
Assume for contradiction that it is finitely accessible. Then it would be ℵ₁-accessible (by <a href="/category-implication/finitely_accessible_raise">this result</a>) – contradiction.


category_id: Set_c
property_id: generalized variety
reason:
Assume for contradiction that it is a generalized variety. Then it would be ℵ₁-accessible (by <a href="/category-implication/generalized_variety_implies_accessible">this result</a>) – contradiction.


category_id: Set_c
property_id: strongly connected
reason:
Assume for contradiction that it is strongly connected. Then it would have an initial object and would be strongly connected and would have a terminal object, so it would be pointed (by <a href="/category-implication/strongly_connected_pointed_criterion">this result</a>) – contradiction.


category_id: Set_c
property_id: cocomplete
reason:
Assume for contradiction that it is cocomplete. Then it would be multi-cocomplete (by <a href="/category-implication/dual_multi-complete_generalize_limits">this result</a>) – contradiction.


category_id: Set_c
property_id: coproducts
reason:
Assume for contradiction that it has coproducts. Then it would have coequalizers and would have coproducts, so it would be cocomplete (by <a href="/category-implication/dual_complete_characterization">this result</a>) – contradiction.


category_id: Set_c
property_id: infinitary distributive
reason:
Assume for contradiction that it is infinitary distributive. Then it would have coproducts (by <a href="/category-implication/infinitary_distributive_assumption">this result</a>) – contradiction.


category_id: Set_c
property_id: copowers
reason:
Assume for contradiction that it has copowers. Then it would have copowers and would be essentially small, so it would be thin (by <a href="/category-implication/dual_freyd_small">this result</a>) – contradiction.


category_id: Set_c
property_id: essentially countable
reason:
Assume for contradiction that it is essentially countable. Then it would have countable copowers and would be essentially countable, so it would be thin (by <a href="/category-implication/dual_freyd_countable">this result</a>) – contradiction.


category_id: Set_c
property_id: self-dual
reason:
Assume for contradiction that it is self-dual. Then it would have countable coproducts and would be self-dual, so it would have countable products (by <a href="/category-implication/self-dual_countable coproducts">this result</a>) – contradiction.


category_id: Fld
property_id: Cauchy complete
reason:
Since it is left cancellative, it is Cauchy complete (by <a href="/category-implication/cauchy_complete_criterion">this result</a>).


category_id: Fld
property_id: coreflexive equalizers
reason:
Since it is left cancellative, it has coreflexive equalizers (by <a href="/category-implication/reflexive_pair_trivial">this result</a>).


category_id: Fld
property_id: reflexive coequalizers
reason:
Since it is left cancellative, it has reflexive coequalizers (by <a href="/category-implication/reflexive_pair_trivial">this result</a>).


category_id: Fld
property_id: pullbacks
reason:
Since it has wide pullbacks, it has pullbacks (by <a href="/category-implication/wide_pullbacks_criterion">this result</a>).


category_id: Fld
property_id: finitely accessible
reason:
Since it is locally finitely multi-presentable, it is finitely accessible (by <a href="/category-implication/locally_finitely_multi-presentable_definition">this result</a>).


category_id: Fld
property_id: ℵ₁-accessible
reason:
Since it is finitely accessible, it is ℵ₁-accessible (by <a href="/category-implication/finitely_accessible_raise">this result</a>).


category_id: Fld
property_id: accessible
reason:
Since it is ℵ₁-accessible, it is accessible (by <a href="/category-implication/countably_accessible_special_case">this result</a>).


category_id: Fld
property_id: filtered colimits
reason:
Since it is finitely accessible, it has filtered colimits (by <a href="/category-implication/accessible_require_filtered_colimit">this result</a>).


category_id: Fld
property_id: sifted colimits
reason:
Since it has filtered colimits and has pullbacks and has reflexive coequalizers, it has sifted colimits (by <a href="/category-implication/sifted_colimits_criterion">this result</a>).


category_id: Fld
property_id: filtered-colimit-stable monomorphisms
reason:
Since it has filtered colimits and is left cancellative, it has filtered-colimit-stable monomorphisms (by <a href="/category-implication/filtered_monos_trivial">this result</a>).


category_id: Fld
property_id: generalized variety
reason:
Since it is multi-algebraic, it is a generalized variety (by <a href="/category-implication/multi-algebraic_another_definition">this result</a>).


category_id: Fld
property_id: sequential limits
reason:
Since it has directed limits, it has sequential limits (by <a href="/category-implication/directed_limits_consequence">this result</a>).


category_id: Fld
property_id: binary products
reason:
Assume for contradiction that it has binary products. Then it would have binary powers (by <a href="/category-implication/binary_products_include_binary_powers">this result</a>) – contradiction.


category_id: Fld
property_id: finite products
reason:
Assume for contradiction that it has finite products. Then it would have binary products (by <a href="/category-implication/finite_products_characterization">this result</a>) – contradiction.


category_id: Fld
property_id: products
reason:
Assume for contradiction that it has products. Then it would have finite products (by <a href="/category-implication/products_consequences">this result</a>) – contradiction.


category_id: Fld
property_id: terminal object
reason:
Assume for contradiction that it has a terminal object. Then it would have pullbacks and would have a terminal object, so it would have binary products (by <a href="/category-implication/binary_products_criterion">this result</a>) – contradiction.


category_id: Fld
property_id: countable products
reason:
Assume for contradiction that it has countable products. Then it would have finite products (by <a href="/category-implication/countable_products_consequence">this result</a>) – contradiction.


category_id: Fld
property_id: finitely complete
reason:
Assume for contradiction that it is finitely complete. Then it would have finite products (by <a href="/category-implication/finitely_complete_characterization">this result</a>) – contradiction.


category_id: Fld
property_id: finite powers
reason:
Assume for contradiction that it has finite powers. Then it would have binary powers (by <a href="/category-implication/finite_powers_consequences">this result</a>) – contradiction.


category_id: Fld
property_id: countable powers
reason:
Assume for contradiction that it has countable powers. Then it would have finite powers (by <a href="/category-implication/countable_powers_include_finite_powers">this result</a>) – contradiction.


category_id: Fld
property_id: multi-complete
reason:
Assume for contradiction that it is multi-complete. Then it would have a multi-terminal object (by <a href="/category-implication/multi-terminal_special_case">this result</a>) – contradiction.


category_id: Fld
property_id: biproducts
reason:
Assume for contradiction that it has biproducts. Then it would have finite products (by <a href="/category-implication/biproducts_definition_consequences">this result</a>) – contradiction.


category_id: Fld
property_id: exact filtered colimits
reason:
Assume for contradiction that it has exact filtered colimits. Then it would be finitely complete (by <a href="/category-implication/exact_filtered_colimits_condition">this result</a>) – contradiction.


category_id: Fld
property_id: cartesian filtered colimits
reason:
Assume for contradiction that it has cartesian filtered colimits. Then it would have finite products (by <a href="/category-implication/cartesian_filtered_colimits_condition">this result</a>) – contradiction.


category_id: Fld
property_id: infinitary distributive
reason:
Assume for contradiction that it is infinitary distributive. Then it would have finite products (by <a href="/category-implication/infinitary_distributive_assumption">this result</a>) – contradiction.


category_id: Fld
property_id: countably distributive
reason:
Assume for contradiction that it is countably distributive. Then it would have finite products (by <a href="/category-implication/countably_distributive_assumption">this result</a>) – contradiction.


category_id: Fld
property_id: distributive
reason:
Assume for contradiction that it is distributive. Then it would have finite products (by <a href="/category-implication/distributive_assumption">this result</a>) – contradiction.


category_id: Fld
property_id: regular
reason:
Assume for contradiction that it is regular. Then it would be finitely complete (by <a href="/category-implication/regular_def">this result</a>) – contradiction.


category_id: Fld
property_id: groupoid
reason:
Assume for contradiction that it is a groupoid. Then it would be locally cartesian closed (by <a href="/category-implication/groupoid_lcc">this result</a>) – contradiction.


category_id: Fld
property_id: thin
reason:
Assume for contradiction that it is thin. Then it would be one-way (by <a href="/category-implication/thin_consequences">this result</a>) – contradiction.


category_id: Fld
property_id: zero morphisms
reason:
Assume for contradiction that it has zero morphisms. Then it would be left cancellative and would have zero morphisms, so it would be thin (by <a href="/category-implication/zero-no-mono">this result</a>) – contradiction.


category_id: Fld
property_id: binary copowers
reason:
Assume for contradiction that it has binary copowers. Then it would have binary copowers and would be left cancellative, so it would be thin (by <a href="/category-implication/codiagonal-no-mono">this result</a>) – contradiction.


category_id: Fld
property_id: gaunt
reason:
Assume for contradiction that it is gaunt. Then it would be core-thin (by <a href="/category-implication/gaunt_characterization">this result</a>) – contradiction.


category_id: Fld
property_id: additive
reason:
Assume for contradiction that it is additive. Then it would have finite products (by <a href="/category-implication/additive_definition">this result</a>) – contradiction.


category_id: Fld
property_id: locally strongly finitely presentable
reason:
Assume for contradiction that it is locally strongly finitely presentable. Then it would be regular (by <a href="/category-implication/many_sorted_algebraic_implies_regular">this result</a>) – contradiction.


category_id: Fld
property_id: cocomplete
reason:
Assume for contradiction that it is cocomplete. Then it would be cocomplete and would be finitely accessible, so it would be locally finitely presentable (by <a href="/category-implication/locally_presentable_definition_finite">this result</a>) – contradiction.


category_id: Fld
property_id: cartesian closed
reason:
Assume for contradiction that it is cartesian closed. Then it would have finite products (by <a href="/category-implication/ccc_condition">this result</a>) – contradiction.


category_id: Fld
property_id: subobject classifier
reason:
Assume for contradiction that it has a subobject classifier. Then it would be finitely complete (by <a href="/category-implication/subobject_classifier_consequence">this result</a>) – contradiction.


category_id: Fld
property_id: regular subobject classifier
reason:
Assume for contradiction that it has a regular subobject classifier. Then it would be finitely complete (by <a href="/category-implication/regular_subobject_assumption">this result</a>) – contradiction.


category_id: Fld
property_id: natural numbers object
reason:
Assume for contradiction that it has a natural numbers object. Then it would have finite products (by <a href="/category-implication/nno_assumption">this result</a>) – contradiction.


category_id: Fld
property_id: semi-strongly connected
reason:
Assume for contradiction that it is semi-strongly connected. Then it would be connected (by <a href="/category-implication/semi-strongly_connected_consequence">this result</a>) – contradiction.


category_id: Fld
property_id: Malcev
reason:
Assume for contradiction that it is Malcev. Then it would be finitely complete (by <a href="/category-implication/malcev_condition">this result</a>) – contradiction.


category_id: Fld
property_id: unital
reason:
Assume for contradiction that it is unital. Then it would be finitely complete (by <a href="/category-implication/unital_assumptions">this result</a>) – contradiction.


category_id: Fld
property_id: sifted
reason:
Assume for contradiction that it is sifted. Then it would be connected (by <a href="/category-implication/sifted_is_connected">this result</a>) – contradiction.


category_id: Fld
property_id: filtered
reason:
Assume for contradiction that it is filtered. Then it would be sifted (by <a href="/category-implication/filtered_is_sifted">this result</a>) – contradiction.


category_id: Fld
property_id: binary coproducts
reason:
Assume for contradiction that it has binary coproducts. Then it would have binary coproducts and would be inhabited, so it would be sifted (by <a href="/category-implication/coproducts_implies_sifted">this result</a>) – contradiction.


category_id: Fld
property_id: finitely cocomplete
reason:
Assume for contradiction that it is finitely cocomplete. Then it would be filtered (by <a href="/category-implication/filtered_criterion">this result</a>) – contradiction.


category_id: Fld
property_id: finite coproducts
reason:
Assume for contradiction that it has finite coproducts. Then it would have binary coproducts (by <a href="/category-implication/dual_finite_products_characterization">this result</a>) – contradiction.


category_id: Fld
property_id: disjoint finite coproducts
reason:
Assume for contradiction that it has disjoint finite coproducts. Then it would have finite coproducts (by <a href="/category-implication/disjoint_coproducts_consequence">this result</a>) – contradiction.


category_id: Fld
property_id: extensive
reason:
Assume for contradiction that it is extensive. Then it would have finite coproducts (by <a href="/category-implication/extensive_assumption">this result</a>) – contradiction.


category_id: Fld
property_id: coproducts
reason:
Assume for contradiction that it has coproducts. Then it would have finite coproducts (by <a href="/category-implication/dual_products_consequences">this result</a>) – contradiction.


category_id: Fld
property_id: countable coproducts
reason:
Assume for contradiction that it has countable coproducts. Then it would have finite coproducts (by <a href="/category-implication/dual_countable_products_consequence">this result</a>) – contradiction.


category_id: Fld
property_id: wide pushouts
reason:
Assume for contradiction that it has wide pushouts. Then it would have pushouts (by <a href="/category-implication/dual_wide_pullbacks_criterion">this result</a>) – contradiction.


category_id: Fld
property_id: connected colimits
reason:
Assume for contradiction that it has connected colimits. Then it would have wide pushouts (by <a href="/category-implication/dual_connected_limits_characterization">this result</a>) – contradiction.


category_id: Fld
property_id: finite copowers
reason:
Assume for contradiction that it has finite copowers. Then it would have binary copowers (by <a href="/category-implication/dual_finite_powers_consequences">this result</a>) – contradiction.


category_id: Fld
property_id: countable copowers
reason:
Assume for contradiction that it has countable copowers. Then it would have finite copowers (by <a href="/category-implication/dual_countable_powers_include_finite_powers">this result</a>) – contradiction.


category_id: Fld
property_id: copowers
reason:
Assume for contradiction that it has copowers. Then it would have countable copowers (by <a href="/category-implication/dual_powers_include_countable_powers">this result</a>) – contradiction.


category_id: Fld
property_id: initial object
reason:
Assume for contradiction that it has an initial object. Then it would be connected (by <a href="/category-implication/dual_multi-terminal_with_connected">this result</a>) – contradiction.


category_id: Fld
property_id: strict initial object
reason:
Assume for contradiction that it has a strict initial object. Then it would have an initial object (by <a href="/category-implication/strict_initial_consequence">this result</a>) – contradiction.


category_id: Fld
property_id: disjoint finite products
reason:
Assume for contradiction that it has disjoint finite products. Then it would have finite products (by <a href="/category-implication/dual_disjoint_coproducts_consequence">this result</a>) – contradiction.


category_id: Fld
property_id: exact cofiltered limits
reason:
Assume for contradiction that it has exact cofiltered limits. Then it would be finitely cocomplete (by <a href="/category-implication/dual_exact_filtered_colimits_condition">this result</a>) – contradiction.


category_id: Fld
property_id: cocartesian cofiltered limits
reason:
Assume for contradiction that it has cocartesian cofiltered limits. Then it would have finite coproducts (by <a href="/category-implication/dual_cartesian_filtered_colimits_condition">this result</a>) – contradiction.


category_id: Fld
property_id: infinitary codistributive
reason:
Assume for contradiction that it is infinitary codistributive. Then it would have finite coproducts (by <a href="/category-implication/dual_infinitary_distributive_assumption">this result</a>) – contradiction.


category_id: Fld
property_id: codistributive
reason:
Assume for contradiction that it is codistributive. Then it would have finite coproducts (by <a href="/category-implication/dual_distributive_assumption">this result</a>) – contradiction.


category_id: Fld
property_id: strict terminal object
reason:
Assume for contradiction that it has a strict terminal object. Then it would have a terminal object (by <a href="/category-implication/dual_strict_initial_consequence">this result</a>) – contradiction.


category_id: Fld
property_id: coextensive
reason:
Assume for contradiction that it is coextensive. Then it would have finite products (by <a href="/category-implication/dual_extensive_assumption">this result</a>) – contradiction.


category_id: Fld
property_id: coregular
reason:
Assume for contradiction that it is coregular. Then it would be finitely cocomplete (by <a href="/category-implication/dual_regular_def">this result</a>) – contradiction.


category_id: Fld
property_id: essentially small
reason:
Assume for contradiction that it is essentially small. Then it would have a cogenerating set (by <a href="/category-implication/dual_essentially_small_consequences">this result</a>) – contradiction.


category_id: Fld
property_id: coequalizers
reason:
Assume for contradiction that it has coequalizers. Then it would have coequalizers and would be left cancellative, so it would be thin (by <a href="/category-implication/dual_cancellative_consequence">this result</a>) – contradiction.


category_id: Fld
property_id: locally copresentable
reason:
Assume for contradiction that it is locally copresentable. Then it would be coaccessible (by <a href="/category-implication/dual_locally_presentable_definition">this result</a>) – contradiction.


category_id: Fld
property_id: cocartesian coclosed
reason:
Assume for contradiction that it is cocartesian coclosed. Then it would have finite coproducts (by <a href="/category-implication/dual_ccc_condition">this result</a>) – contradiction.


category_id: Fld
property_id: regular quotient object classifier
reason:
Assume for contradiction that it has a regular quotient object classifier. Then it would be finitely cocomplete (by <a href="/category-implication/dual_regular_subobject_assumption">this result</a>) – contradiction.


category_id: Fld
property_id: locally cocartesian coclosed
reason:
Assume for contradiction that it is locally cocartesian coclosed. Then it would have pushouts (by <a href="/category-implication/dual_pullbacks_are_local_products">this result</a>) – contradiction.


category_id: Fld
property_id: co-Malcev
reason:
Assume for contradiction that it is co-Malcev. Then it would be finitely cocomplete (by <a href="/category-implication/dual_malcev_condition">this result</a>) – contradiction.


category_id: Fld
property_id: counital
reason:
Assume for contradiction that it is counital. Then it would be finitely cocomplete (by <a href="/category-implication/dual_unital_assumptions">this result</a>) – contradiction.


category_id: Fld
property_id: cosifted
reason:
Assume for contradiction that it is cosifted. Then it would be connected (by <a href="/category-implication/dual_sifted_is_connected">this result</a>) – contradiction.


category_id: Fld
property_id: cofiltered
reason:
Assume for contradiction that it is cofiltered. Then it would be cosifted (by <a href="/category-implication/dual_filtered_is_sifted">this result</a>) – contradiction.


category_id: Fld
property_id: self-dual
reason:
Assume for contradiction that it is self-dual. Then it would have pullbacks and would be self-dual, so it would have pushouts (by <a href="/category-implication/self-dual_pullbacks">this result</a>) – contradiction.


category_id: FinAb
property_id: essentially small
reason:
Since it is essentially countable, it is essentially small (by <a href="/category-implication/essentially_countable_consequence">this result</a>).


category_id: FinAb
property_id: coequalizers
reason:
Since it has cokernels and is preadditive, it has coequalizers (by <a href="/category-implication/dual_equalizers_via_kernels">this result</a>).


category_id: FinAb
property_id: finitely cocomplete
reason:
Since it has coequalizers and has finite coproducts, it is finitely cocomplete (by <a href="/category-implication/dual_finitely_complete_characterization">this result</a>).


category_id: FinAb
property_id: finite
reason:
Assume for contradiction that it is finite. Then it would be countable (by <a href="/category-implication/finite_consequence">this result</a>) – contradiction.


category_id: FinAb
property_id: coproducts
reason:
Assume for contradiction that it has coproducts. Then it would have coproducts and would have a generating set and would have zero morphisms, so it would have a generator (by <a href="/category-implication/generator_via_coproduct">this result</a>) – contradiction.


category_id: FinAb
property_id: Grothendieck abelian
reason:
Assume for contradiction that it is Grothendieck abelian. Then it would have coproducts (by <a href="/category-implication/grothendieck_abelian_definition">this result</a>) – contradiction.


category_id: FinAb
property_id: cocomplete
reason:
Assume for contradiction that it is cocomplete. Then it would have connected colimits (by <a href="/category-implication/dual_complete_consequence">this result</a>) – contradiction.


category_id: FinAb
property_id: countable copowers
reason:
Assume for contradiction that it has countable copowers. Then it would have countable copowers and would be essentially countable, so it would be thin (by <a href="/category-implication/dual_freyd_countable">this result</a>) – contradiction.


category_id: FinAb
property_id: countable coproducts
reason:
Assume for contradiction that it has countable coproducts. Then it would have countable copowers (by <a href="/category-implication/dual_countable_products_consequence">this result</a>) – contradiction.


category_id: FinAb
property_id: cogenerator
reason:
Assume for contradiction that it has a cogenerator. Then it would have a cogenerator and would be self-dual, so it would have a generator (by <a href="/category-implication/self-dual_cogenerator">this result</a>) – contradiction.


category_id: FinGrp
property_id: unital
reason:
Since it is Malcev and is pointed, it is unital (by <a href="/category-implication/malcev_implies_unital">this result</a>).


category_id: FinGrp
property_id: thin
reason:
Assume for contradiction that it is thin. Then it would be left cancellative (by <a href="/category-implication/thin_consequence">this result</a>) – contradiction.


category_id: FinGrp
property_id: cartesian closed
reason:
Assume for contradiction that it is cartesian closed. Then it would be cartesian closed and would have an initial object, so it would have a strict initial object (by <a href="/category-implication/ccc_consequence">this result</a>) – contradiction.


category_id: FinGrp
property_id: subobject classifier
reason:
Assume for contradiction that it has a subobject classifier. Then it would be pointed and would have a subobject classifier, so it would be normal (by <a href="/category-implication/subobject_classifier_pointed_case">this result</a>) – contradiction.


category_id: FinGrp
property_id: regular subobject classifier
reason:
Assume for contradiction that it has a regular subobject classifier. Then it would be mono-regular and would have a regular subobject classifier, so it would have a subobject classifier (by <a href="/category-implication/regular_subobjects_suffice">this result</a>) – contradiction.


category_id: FinGrp
property_id: directed colimits
reason:
Assume for contradiction that it has directed colimits. Then it would have sequential colimits (by <a href="/category-implication/dual_directed_limits_consequence">this result</a>) – contradiction.


category_id: FinGrp
property_id: filtered colimits
reason:
Assume for contradiction that it has filtered colimits. Then it would have directed colimits (by <a href="/category-implication/directed_colimits_suffice">this result</a>) – contradiction.


category_id: FinGrp
property_id: countable coproducts
reason:
Assume for contradiction that it has countable coproducts. Then it would have coequalizers and would have countable coproducts, so it would have sequential colimits (by <a href="/category-implication/dual_sequential_limits_criterion">this result</a>) – contradiction.


category_id: FinGrp
property_id: binary coproducts
reason:
Assume for contradiction that it has binary coproducts. Then it would have binary copowers (by <a href="/category-implication/dual_binary_products_include_binary_powers">this result</a>) – contradiction.


category_id: FinGrp
property_id: biproducts
reason:
Assume for contradiction that it has biproducts. Then it would have finite coproducts (by <a href="/category-implication/biproducts_definition_consequences">this result</a>) – contradiction.


category_id: FinGrp
property_id: pushouts
reason:
Assume for contradiction that it has pushouts. Then it would have an initial object and would have pushouts, so it would have binary coproducts (by <a href="/category-implication/dual_binary_products_criterion">this result</a>) – contradiction.


category_id: FinGrp
property_id: finitely cocomplete
reason:
Assume for contradiction that it is finitely cocomplete. Then it would have finite coproducts (by <a href="/category-implication/dual_finitely_complete_characterization">this result</a>) – contradiction.


category_id: FinGrp
property_id: strict terminal object
reason:
Assume for contradiction that it has a strict terminal object. Then it would be pointed and would have a strict terminal object, so it would be trivial (by <a href="/category-implication/dual_strict_initial_trivial">this result</a>) – contradiction.


category_id: FinGrp
property_id: quotient object classifier
reason:
Assume for contradiction that it has a quotient object classifier. Then it would be finitely cocomplete (by <a href="/category-implication/dual_subobject_classifier_consequence">this result</a>) – contradiction.


category_id: FinOrd
property_id: quotients of congruences
reason:
Since it is core-thin, it has quotients of congruences (by <a href="/category-implication/core-hin_quotients">this result</a>).


category_id: FinOrd
property_id: cosifted
reason:
Since it is cofiltered, it is cosifted (by <a href="/category-implication/dual_filtered_is_sifted">this result</a>).


category_id: FinOrd
property_id: countable products
reason:
Assume for contradiction that it has countable products. Then it would have countable products and would have equalizers, so it would have sequential limits (by <a href="/category-implication/sequential_limits_criterion">this result</a>) – contradiction.


category_id: FinOrd
property_id: pullbacks
reason:
Assume for contradiction that it has pullbacks. Then it would have pullbacks and would have a terminal object, so it would have binary products (by <a href="/category-implication/binary_products_criterion">this result</a>) – contradiction.


category_id: FinOrd
property_id: direct
reason:
Assume for contradiction that it is direct. Then it would be one-way (by <a href="/category-implication/direct_implies_one-way">this result</a>) – contradiction.


category_id: FinOrd
property_id: locally cartesian closed
reason:
Assume for contradiction that it is locally cartesian closed. Then it would have pullbacks (by <a href="/category-implication/pullbacks_are_local_products">this result</a>) – contradiction.


category_id: FinOrd
property_id: left cancellative
reason:
Assume for contradiction that it is left cancellative. Then it would be left cancellative and would be sifted, so it would be thin (by <a href="/category-implication/sifted_left_cancellative_implies_thin">this result</a>) – contradiction.


category_id: FinOrd
property_id: cofiltered limits
reason:
Assume for contradiction that it has cofiltered limits. Then it would have directed limits (by <a href="/category-implication/dual_directed_colimits_suffice">this result</a>) – contradiction.


category_id: B
property_id: directed limits
reason:
Since it is a groupoid, it has directed limits (by <a href="/category-implication/groupoid_consequence">this result</a>).


category_id: B
property_id: left cancellative
reason:
Since it is a groupoid, it is left cancellative (by <a href="/category-implication/groupoid_consequence">this result</a>).


category_id: B
property_id: mono-regular
reason:
Since it is a groupoid, it is mono-regular (by <a href="/category-implication/groupoid_consequence">this result</a>).


category_id: B
property_id: pullbacks
reason:
Since it is a groupoid, it has pullbacks (by <a href="/category-implication/groupoid_consequence">this result</a>).


category_id: B
property_id: self-dual
reason:
Since it is a groupoid, it is self-dual (by <a href="/category-implication/groupoid_consequence">this result</a>).


category_id: B
property_id: locally cartesian closed
reason:
Since it is a groupoid, it is locally cartesian closed (by <a href="/category-implication/groupoid_lcc">this result</a>).


category_id: B
property_id: cofiltered limits
reason:
Since it has directed limits, it has cofiltered limits (by <a href="/category-implication/dual_directed_colimits_suffice">this result</a>).


category_id: B
property_id: wide pullbacks
reason:
Since it has cofiltered limits and has pullbacks, it has wide pullbacks (by <a href="/category-implication/wide_pullbacks_criterion">this result</a>).


category_id: B
property_id: directed colimits
reason:
Since it is a groupoid, it has directed colimits (by <a href="/category-implication/dual_groupoid_consequence">this result</a>).


category_id: B
property_id: filtered colimits
reason:
Since it has directed colimits, it has filtered colimits (by <a href="/category-implication/directed_colimits_suffice">this result</a>).


category_id: B
property_id: epi-regular
reason:
Since it is a groupoid, it is epi-regular (by <a href="/category-implication/dual_groupoid_consequence">this result</a>).


category_id: B
property_id: pushouts
reason:
Since it is a groupoid, it has pushouts (by <a href="/category-implication/dual_groupoid_consequence">this result</a>).


category_id: B
property_id: cosifted limits
reason:
Since it has cofiltered limits and has coreflexive equalizers and has pushouts, it has cosifted limits (by <a href="/category-implication/dual_sifted_colimits_criterion">this result</a>).


category_id: B
property_id: wide pushouts
reason:
Since it has filtered colimits and has pushouts, it has wide pushouts (by <a href="/category-implication/dual_wide_pullbacks_criterion">this result</a>).


category_id: B
property_id: right cancellative
reason:
Since it is a groupoid, it is right cancellative (by <a href="/category-implication/dual_groupoid_consequence">this result</a>).


category_id: B
property_id: locally cocartesian coclosed
reason:
Since it is a groupoid, it is locally cocartesian coclosed (by <a href="/category-implication/dual_groupoid_lcc">this result</a>).


category_id: B
property_id: cofiltered-limit-stable epimorphisms
reason:
Since it has cofiltered limits and is right cancellative, it has cofiltered-limit-stable epimorphisms (by <a href="/category-implication/dual_filtered_monos_trivial">this result</a>).


category_id: B
property_id: terminal object
reason:
Assume for contradiction that it has a terminal object. Then it would be connected (by <a href="/category-implication/multi-terminal_with_connected">this result</a>) – contradiction.


category_id: B
property_id: finite products
reason:
Assume for contradiction that it has finite products. Then it would have a terminal object (by <a href="/category-implication/finite_products_characterization">this result</a>) – contradiction.


category_id: B
property_id: finite powers
reason:
Assume for contradiction that it has finite powers. Then it would have a terminal object (by <a href="/category-implication/finite_powers_consequences">this result</a>) – contradiction.


category_id: B
property_id: trivial
reason:
Assume for contradiction that it is trivial. Then it would be essentially finite (by <a href="/category-implication/trivial_consequence">this result</a>) – contradiction.


category_id: B
property_id: thin
reason:
Assume for contradiction that it is thin. Then it would be inhabited and would be thin, so it would have a generator (by <a href="/category-implication/thin_inhabited_consequence">this result</a>) – contradiction.


category_id: B
property_id: equalizers
reason:
Assume for contradiction that it has equalizers. Then it would have equalizers and would be right cancellative, so it would be thin (by <a href="/category-implication/cancellative_consequence">this result</a>) – contradiction.


category_id: B
property_id: binary products
reason:
Assume for contradiction that it has binary products. Then it would have binary products and would have coreflexive equalizers, so it would have equalizers (by <a href="/category-implication/equalizers_via_coreflexive_equalizers">this result</a>) – contradiction.


category_id: B
property_id: connected limits
reason:
Assume for contradiction that it has connected limits. Then it would have equalizers (by <a href="/category-implication/connected_limits_characterization">this result</a>) – contradiction.


category_id: B
property_id: core-thin
reason:
Assume for contradiction that it is core-thin. Then it would be core-thin and would be a groupoid, so it would be thin (by <a href="/category-implication/thin_groupoids">this result</a>) – contradiction.


category_id: B
property_id: essentially discrete
reason:
Assume for contradiction that it is essentially discrete. Then it would be thin (by <a href="/category-implication/essentially_discrete_characterization">this result</a>) – contradiction.


category_id: B
property_id: multi-terminal object
reason:
Assume for contradiction that it has a multi-terminal object. Then it would be a groupoid and would have a multi-terminal object, so it would be thin (by <a href="/category-implication/groupoid_with_multi-terminal">this result</a>) – contradiction.


category_id: B
property_id: cocomplete
reason:
Assume for contradiction that it is cocomplete. Then it would be finitely cocomplete (by <a href="/category-implication/dual_complete_consequence">this result</a>) – contradiction.


category_id: B
property_id: initial object
reason:
Assume for contradiction that it has an initial object. Then it would have an initial object and would have pushouts, so it would have binary coproducts (by <a href="/category-implication/dual_binary_products_criterion">this result</a>) – contradiction.


category_id: B
property_id: connected colimits
reason:
Assume for contradiction that it has connected colimits. Then it would have coequalizers (by <a href="/category-implication/dual_connected_limits_characterization">this result</a>) – contradiction.


category_id: B
property_id: binary powers
reason:
Assume for contradiction that it has binary powers. Then it would have binary powers and would be right cancellative, so it would be thin (by <a href="/category-implication/dual_codiagonal-no-mono">this result</a>) – contradiction.


category_id: B
property_id: multi-initial object
reason:
Assume for contradiction that it has a multi-initial object. Then it would be a groupoid and would have a multi-initial object, so it would be thin (by <a href="/category-implication/dual_groupoid_with_multi-terminal">this result</a>) – contradiction.


category_id: FI
property_id: connected limits
reason:
Since it has equalizers and has wide pullbacks, it has connected limits (by <a href="/category-implication/connected_limits_characterization">this result</a>).


category_id: FI
property_id: strict initial object
reason:
Since it has an initial object and is left cancellative, it has a strict initial object (by <a href="/category-implication/strict_initial_left_criterion">this result</a>).


category_id: FI
property_id: connected
reason:
Since it is semi-strongly connected, it is connected (by <a href="/category-implication/semi-strongly_connected_consequence">this result</a>).


category_id: FI
property_id: multi-terminal object
reason:
Assume for contradiction that it has a multi-terminal object. Then it would be connected and would have a multi-terminal object, so it would have a terminal object (by <a href="/category-implication/multi-terminal_with_connected">this result</a>) – contradiction.


category_id: FI
property_id: groupoid
reason:
Assume for contradiction that it is a groupoid. Then it would be connected and would be a groupoid, so it would be strongly connected (by <a href="/category-implication/groupoid_connected">this result</a>) – contradiction.


category_id: FI
property_id: right cancellative
reason:
Assume for contradiction that it is right cancellative. Then it would be balanced and would be left cancellative and would be right cancellative, so it would be a groupoid (by <a href="/category-implication/groupoid_criterion">this result</a>) – contradiction.


category_id: FI
property_id: sifted
reason:
Assume for contradiction that it is sifted. Then it would be left cancellative and would be sifted, so it would be thin (by <a href="/category-implication/sifted_left_cancellative_implies_thin">this result</a>) – contradiction.


category_id: FI
property_id: coequalizers
reason:
Assume for contradiction that it has coequalizers. Then it would have coequalizers and would be semi-strongly connected, so it would be filtered (by <a href="/category-implication/filtered_via_equalizers">this result</a>) – contradiction.


category_id: FI
property_id: wide pushouts
reason:
Assume for contradiction that it has wide pushouts. Then it would have an initial object and would have wide pushouts, so it would be cocomplete (by <a href="/category-implication/dual_complete_criterion">this result</a>) – contradiction.


category_id: FS
property_id: filtered colimits
reason:
Since it has filtered-colimit-stable monomorphisms, it has filtered colimits (by <a href="/category-implication/filtered_monos_assumption">this result</a>).


category_id: FS
property_id: Cauchy complete
reason:
Since it has sequential colimits, it is Cauchy complete (by <a href="/category-implication/sequential_colimits_consequence">this result</a>).


category_id: FS
property_id: connected colimits
reason:
Since it has coequalizers and has wide pushouts, it has connected colimits (by <a href="/category-implication/dual_connected_limits_characterization">this result</a>).


category_id: FS
property_id: pushouts
reason:
Since it has wide pushouts, it has pushouts (by <a href="/category-implication/dual_wide_pullbacks_criterion">this result</a>).


category_id: FS
property_id: inhabited
reason:
Since it has a cogenerator, it is inhabited (by <a href="/category-implication/dual_generator_consequence">this result</a>).


category_id: FS
property_id: coreflexive equalizers
reason:
Since it is right cancellative, it has coreflexive equalizers (by <a href="/category-implication/dual_reflexive_pair_trivial">this result</a>).


category_id: FS
property_id: wide pullbacks
reason:
Assume for contradiction that it has wide pullbacks. Then it would have pullbacks (by <a href="/category-implication/wide_pullbacks_criterion">this result</a>) – contradiction.


category_id: FS
property_id: complete
reason:
Assume for contradiction that it is complete. Then it would have connected limits (by <a href="/category-implication/complete_consequence">this result</a>) – contradiction.


category_id: FS
property_id: left cancellative
reason:
Assume for contradiction that it is left cancellative. Then it would be balanced and would be left cancellative and would be right cancellative, so it would be a groupoid (by <a href="/category-implication/groupoid_criterion">this result</a>) – contradiction.


category_id: FS
property_id: self-dual
reason:
Assume for contradiction that it is self-dual. Then it would have pushouts and would be self-dual, so it would have pullbacks (by <a href="/category-implication/self-dual_pushouts">this result</a>) – contradiction.


category_id: Ab_fg
property_id: Grothendieck abelian
reason:
Assume for contradiction that it is Grothendieck abelian. Then it would have a cogenerator (by <a href="/category-implication/grothendieck_abelian_cogenerator">this result</a>) – contradiction.


category_id: Ab_fg
property_id: coproducts
reason:
Assume for contradiction that it has coproducts. Then it would have countable coproducts (by <a href="/category-implication/dual_products_consequences">this result</a>) – contradiction.


category_id: Ab_fg
property_id: self-dual
reason:
Assume for contradiction that it is self-dual. Then it would have a generator and would be self-dual, so it would have a cogenerator (by <a href="/category-implication/self-dual_generator">this result</a>) – contradiction.


category_id: FreeAb
property_id: cartesian closed
reason:
Assume for contradiction that it is cartesian closed. Then it would be cartesian closed and would have copowers, so it would have powers (by <a href="/category-implication/power_construction">this result</a>) – contradiction.


category_id: FreeAb
property_id: coequalizers
reason:
Assume for contradiction that it has coequalizers. Then it would have coequalizers and would have coproducts, so it would be cocomplete (by <a href="/category-implication/dual_complete_characterization">this result</a>) – contradiction.


category_id: FreeAb
property_id: reflexive coequalizers
reason:
Assume for contradiction that it has reflexive coequalizers. Then it would have binary coproducts and would have reflexive coequalizers, so it would have coequalizers (by <a href="/category-implication/dual_equalizers_via_coreflexive_equalizers">this result</a>) – contradiction.


category_id: FreeAb
property_id: finitely cocomplete
reason:
Assume for contradiction that it is finitely cocomplete. Then it would have coequalizers (by <a href="/category-implication/dual_finitely_complete_characterization">this result</a>) – contradiction.


category_id: FreeAb
property_id: pushouts
reason:
Assume for contradiction that it has pushouts. Then it would have binary coproducts and would have pushouts, so it would have coequalizers (by <a href="/category-implication/dual_equalizers_criterion">this result</a>) – contradiction.


category_id: FreeAb
property_id: cokernels
reason:
Assume for contradiction that it has cokernels. Then it would have cokernels and would be preadditive, so it would have coequalizers (by <a href="/category-implication/dual_equalizers_via_kernels">this result</a>) – contradiction.


category_id: FreeAb
property_id: quotients of congruences
reason:
Assume for contradiction that it has quotients of congruences. Then it would be preadditive and would have quotients of congruences and would be regular, so it would have cokernels (by <a href="/category-implication/cokernels_via_congruence_quotients">this result</a>) – contradiction.


category_id: FreeAb
property_id: multi-cocomplete
reason:
Assume for contradiction that it is multi-cocomplete. Then it would have finite products and would be multi-cocomplete, so it would be cocomplete (by <a href="/category-implication/dual_multi-complete_with_finite_coproducts">this result</a>) – contradiction.


category_id: FreeAb
property_id: CSP
reason:
Assume for contradiction that it satisfies CSP. Then it would have products (by <a href="/category-implication/dual_CIP_assumption">this result</a>) – contradiction.


category_id: FreeAb
property_id: essentially small
reason:
Assume for contradiction that it is essentially small. Then it would have copowers and would be essentially small, so it would be thin (by <a href="/category-implication/dual_freyd_small">this result</a>) – contradiction.


category_id: Grp
property_id: biproducts
reason:
Assume for contradiction that it has biproducts. Then it would have biproducts and would have filtered colimits and would have filtered-colimit-stable monomorphisms and would have products, so it would satisfy CIP (by <a href="/category-implication/CIP_criterion">this result</a>) – contradiction.


category_id: Grp
property_id: co-Malcev
reason:
Assume for contradiction that it is co-Malcev. Then it would be co-Malcev and would be pointed, so it would be counital (by <a href="/category-implication/dual_malcev_implies_unital">this result</a>) – contradiction.


category_id: Haus
property_id: finite products
reason:
Since it has products, it has finite products (by <a href="/category-implication/products_consequences">this result</a>).


category_id: Haus
property_id: connected limits
reason:
Since it is complete, it has connected limits (by <a href="/category-implication/complete_consequence">this result</a>).


category_id: Haus
property_id: multi-terminal object
reason:
Since it is multi-complete, it has a multi-terminal object (by <a href="/category-implication/multi-terminal_special_case">this result</a>).


category_id: Haus
property_id: coproducts
reason:
Since it is infinitary extensive, it has coproducts (by <a href="/category-implication/infinitary_extensive_assumption">this result</a>).


category_id: Haus
property_id: extensive
reason:
Since it is infinitary extensive, it is extensive (by <a href="/category-implication/infinitary_extensive_finitary">this result</a>).


category_id: Haus
property_id: finite coproducts
reason:
Since it is extensive, it has finite coproducts (by <a href="/category-implication/extensive_assumption">this result</a>).


category_id: Haus
property_id: cocartesian cofiltered limits
reason:
Since it has cofiltered limits and is extensive and has a terminal object, it has cocartesian cofiltered limits (by <a href="/category-implication/extensive_cocartesian_cofiltered_limits">this result</a>).


category_id: Haus
property_id: strict initial object
reason:
Since it is extensive, it has a strict initial object (by <a href="/category-implication/extensive_consequences">this result</a>).


category_id: Haus
property_id: distributive
reason:
Since it is extensive and has finite products, it is distributive (by <a href="/category-implication/extensive_distributivity">this result</a>).


category_id: Haus
property_id: infinitary distributive
reason:
Since it has finite products and is infinitary extensive, it is infinitary distributive (by <a href="/category-implication/infinitary_extensive_distributivity">this result</a>).


category_id: Haus
property_id: countably distributive
reason:
Since it is infinitary distributive, it is countably distributive (by <a href="/category-implication/infinitary_distributive_trivial">this result</a>).


category_id: Haus
property_id: filtered colimits
reason:
Since it has sifted colimits, it has filtered colimits (by <a href="/category-implication/sifted_colimits_consequence">this result</a>).


category_id: Haus
property_id: biproducts
reason:
Assume for contradiction that it has biproducts. Then it would have biproducts and would have filtered colimits, so it would have cartesian filtered colimits (by <a href="/category-implication/biproducts_cartesian_filtered_colimits">this result</a>) – contradiction.


category_id: Haus
property_id: cartesian closed
reason:
Assume for contradiction that it is cartesian closed. Then it would be cartesian closed and would have filtered colimits, so it would have cartesian filtered colimits (by <a href="/category-implication/ccc_cartesian_filtered_colimits">this result</a>) – contradiction.


category_id: Haus
property_id: disjoint products
reason:
Assume for contradiction that it has disjoint products. Then it would have disjoint finite products (by <a href="/category-implication/dual_disjoint_coproducts_equivalence">this result</a>) – contradiction.


category_id: Haus
property_id: coextensive
reason:
Assume for contradiction that it is coextensive. Then it would be coextensive and would have filtered colimits and would have an initial object, so it would have cartesian filtered colimits (by <a href="/category-implication/dual_extensive_cocartesian_cofiltered_limits">this result</a>) – contradiction.


category_id: R-Mod_div
property_id: abelian
reason:
Since it is split abelian, it is abelian (by <a href="/category-implication/split_abelian_condition">this result</a>).


category_id: R-Mod_div
property_id: inverse
reason:
Assume for contradiction that it is inverse. Then it would be skeletal (by <a href="/category-implication/dual_direct_implies_one-way">this result</a>) – contradiction.


category_id: R-Mod_div
property_id: right cancellative
reason:
Assume for contradiction that it is right cancellative. Then it would have cofiltered limits and would be right cancellative, so it would have cofiltered-limit-stable epimorphisms (by <a href="/category-implication/dual_filtered_monos_trivial">this result</a>) – contradiction.


category_id: R-Mod_div
property_id: groupoid
reason:
Assume for contradiction that it is a groupoid. Then it would be right cancellative (by <a href="/category-implication/dual_groupoid_consequence">this result</a>) – contradiction.


category_id: LRS
property_id: equalizers
reason:
Since it is complete, it has equalizers (by <a href="/category-implication/complete_characterization">this result</a>).


category_id: M-Set
property_id: coproducts
reason:
Since it is a Grothendieck topos, it has coproducts (by <a href="/category-implication/grothendieck_topos_definition">this result</a>).


category_id: M-Set
property_id: elementary topos
reason:
Since it is a Grothendieck topos, it is an elementary topos (by <a href="/category-implication/grothendieck_topos_definition">this result</a>).


category_id: M-Set
property_id: infinitary distributive
reason:
Since it is cartesian closed and has coproducts, it is infinitary distributive (by <a href="/category-implication/infinitary_distributive_criterion">this result</a>).


category_id: M-Set
property_id: cogenerator
reason:
Since it is a Grothendieck topos, it has a cogenerator (by <a href="/category-implication/grothendieck_topos_consequence">this result</a>).


category_id: M-Set
property_id: infinitary extensive
reason:
Since it is a Grothendieck topos, it is infinitary extensive (by <a href="/category-implication/grothendieck_topos_consequence">this result</a>).


category_id: M-Set
property_id: thin
reason:
Assume for contradiction that it is thin. Then it would be finitely complete and would be thin, so it would be Malcev (by <a href="/category-implication/malcev_thin_condition">this result</a>) – contradiction.


category_id: Meas
property_id: exact filtered colimits
reason:
Assume for contradiction that it has exact filtered colimits. Then it would have cartesian filtered colimits (by <a href="/category-implication/exact_includes_cartesian_filtered_colimits">this result</a>) – contradiction.


category_id: Met_c
property_id: finite products
reason:
Since it has countable products, it has finite products (by <a href="/category-implication/countable_products_consequence">this result</a>).


category_id: Met_c
property_id: reflexive coequalizers
reason:
Assume for contradiction that it has reflexive coequalizers. Then it would have quotients of congruences (by <a href="/category-implication/congruence_quotients_are_reflexive_coequalizers">this result</a>) – contradiction.


category_id: Met_c
property_id: sifted colimits
reason:
Assume for contradiction that it has sifted colimits. Then it would have reflexive coequalizers (by <a href="/category-implication/sifted_colimits_consequence">this result</a>) – contradiction.


category_id: Met_c
property_id: products
reason:
Assume for contradiction that it has products. Then it would have powers (by <a href="/category-implication/products_consequences">this result</a>) – contradiction.


category_id: Met_c
property_id: left cancellative
reason:
Assume for contradiction that it is left cancellative. Then it would have reflexive coequalizers (by <a href="/category-implication/reflexive_pair_trivial">this result</a>) – contradiction.


category_id: Met_c
property_id: one-way
reason:
Assume for contradiction that it is one-way. Then it would have reflexive coequalizers (by <a href="/category-implication/one-way_reflexive">this result</a>) – contradiction.


category_id: Met_c
property_id: essentially countable
reason:
Assume for contradiction that it is essentially countable. Then it would have countable powers and would be essentially countable, so it would be thin (by <a href="/category-implication/freyd_countable">this result</a>) – contradiction.


category_id: Met_c
property_id: locally strongly finitely presentable
reason:
Assume for contradiction that it is locally strongly finitely presentable. Then it would be a generalized variety (by <a href="/category-implication/locally_strongly_finitely_presentable_definition">this result</a>) – contradiction.


category_id: Met_c
property_id: coequalizers
reason:
Assume for contradiction that it has coequalizers. Then it would have reflexive coequalizers (by <a href="/category-implication/dual_coreflexive_equalizers_are_equalizers">this result</a>) – contradiction.


category_id: Met_c
property_id: directed limits
reason:
Assume for contradiction that it has directed limits. Then it would have cofiltered limits (by <a href="/category-implication/dual_directed_colimits_suffice">this result</a>) – contradiction.


category_id: Met
property_id: finite products
reason:
Since it has binary products and has a terminal object, it has finite products (by <a href="/category-implication/finite_products_characterization">this result</a>).


category_id: Met
property_id: infinitary extensive
reason:
Assume for contradiction that it is infinitary extensive. Then it would have finite products and would be infinitary extensive, so it would be infinitary distributive (by <a href="/category-implication/infinitary_extensive_distributivity">this result</a>) – contradiction.


category_id: Met
property_id: distributive
reason:
Assume for contradiction that it is distributive. Then it would have finite coproducts (by <a href="/category-implication/distributive_assumption">this result</a>) – contradiction.


category_id: Met
property_id: cocomplete
reason:
Assume for contradiction that it is cocomplete. Then it would have coproducts (by <a href="/category-implication/dual_complete_characterization">this result</a>) – contradiction.


category_id: Met_oo
property_id: infinitary distributive
reason:
Since it has cartesian filtered colimits and has coproducts and is distributive, it is infinitary distributive (by <a href="/category-implication/infinite_distributive_filtered_criterion">this result</a>).


category_id: Setne
property_id: cartesian filtered colimits
reason:
Since it is cartesian closed and has filtered colimits, it has cartesian filtered colimits (by <a href="/category-implication/ccc_cartesian_filtered_colimits">this result</a>).


category_id: Setne
property_id: sifted
reason:
Since it has binary coproducts and is inhabited, it is sifted (by <a href="/category-implication/coproducts_implies_sifted">this result</a>).


category_id: Setne
property_id: equalizers
reason:
Assume for contradiction that it has equalizers. Then it would have countable products and would have equalizers, so it would have sequential limits (by <a href="/category-implication/sequential_limits_criterion">this result</a>) – contradiction.


category_id: Setne
property_id: coreflexive equalizers
reason:
Assume for contradiction that it has coreflexive equalizers. Then it would have binary products and would have coreflexive equalizers, so it would have equalizers (by <a href="/category-implication/equalizers_via_coreflexive_equalizers">this result</a>) – contradiction.


category_id: Setne
property_id: complete
reason:
Assume for contradiction that it is complete. Then it would have equalizers (by <a href="/category-implication/complete_characterization">this result</a>) – contradiction.


category_id: Setne
property_id: finitely complete
reason:
Assume for contradiction that it is finitely complete. Then it would have equalizers (by <a href="/category-implication/finitely_complete_characterization">this result</a>) – contradiction.


category_id: Setne
property_id: pullbacks
reason:
Assume for contradiction that it has pullbacks. Then it would have binary products and would have pullbacks, so it would have equalizers (by <a href="/category-implication/equalizers_criterion">this result</a>) – contradiction.


category_id: Setne
property_id: finite coproducts
reason:
Assume for contradiction that it has finite coproducts. Then it would have finite coproducts and would be multi-complete, so it would be complete (by <a href="/category-implication/multi-complete_with_finite_coproducts">this result</a>) – contradiction.


category_id: Setne
property_id: CIP
reason:
Assume for contradiction that it satisfies CIP. Then it would have coproducts (by <a href="/category-implication/CIP_assumption">this result</a>) – contradiction.


category_id: Setne
property_id: left cancellative
reason:
Assume for contradiction that it is left cancellative. Then it would have coreflexive equalizers (by <a href="/category-implication/reflexive_pair_trivial">this result</a>) – contradiction.


category_id: Setne
property_id: thin
reason:
Assume for contradiction that it is thin. Then it would have equalizers (by <a href="/category-implication/thin_consequence">this result</a>) – contradiction.


category_id: Setne
property_id: strict terminal object
reason:
Assume for contradiction that it has a strict terminal object. Then it would be cartesian closed and would have a strict terminal object, so it would be thin (by <a href="/category-implication/ccc_no_strict_terminal">this result</a>) – contradiction.


category_id: Setne
property_id: elementary topos
reason:
Assume for contradiction that it is an elementary topos. Then it would be finitely complete (by <a href="/category-implication/topos_definition">this result</a>) – contradiction.


category_id: Setne
property_id: pointed
reason:
Assume for contradiction that it is pointed. Then it would have a natural numbers object and would be pointed, so it would be trivial (by <a href="/category-implication/nno_pointed_case">this result</a>) – contradiction.


category_id: Setne
property_id: initial object
reason:
Assume for contradiction that it has an initial object. Then it would have an initial object and would be strongly connected and would have a terminal object, so it would be pointed (by <a href="/category-implication/strongly_connected_pointed_criterion">this result</a>) – contradiction.


category_id: Setne
property_id: cosifted limits
reason:
Assume for contradiction that it has cosifted limits. Then it would have coreflexive equalizers (by <a href="/category-implication/dual_sifted_colimits_consequence">this result</a>) – contradiction.


category_id: Setne
property_id: finite copowers
reason:
Assume for contradiction that it has finite copowers. Then it would have an initial object (by <a href="/category-implication/dual_finite_powers_consequences">this result</a>) – contradiction.


category_id: Setne
property_id: multi-initial object
reason:
Assume for contradiction that it has a multi-initial object. Then it would be connected and would have a multi-initial object, so it would have an initial object (by <a href="/category-implication/dual_multi-terminal_with_connected">this result</a>) – contradiction.


category_id: Setne
property_id: zero morphisms
reason:
Assume for contradiction that it has zero morphisms. Then it would have a terminal object and would have zero morphisms, so it would be pointed (by <a href="/category-implication/dual_pointed_characterization">this result</a>) – contradiction.


category_id: Setne
property_id: countably codistributive
reason:
Assume for contradiction that it is countably codistributive. Then it would have finite coproducts (by <a href="/category-implication/dual_countably_distributive_assumption">this result</a>) – contradiction.


category_id: Setne
property_id: right cancellative
reason:
Assume for contradiction that it is right cancellative. Then it would be right cancellative and would have a terminal object, so it would have a strict terminal object (by <a href="/category-implication/dual_strict_initial_left_criterion">this result</a>) – contradiction.


category_id: Setne
property_id: locally cocartesian coclosed
reason:
Assume for contradiction that it is locally cocartesian coclosed. Then it would have disjoint finite products and would be locally cocartesian coclosed, so it would be coextensive (by <a href="/category-implication/dual_lcc_implies_extensive">this result</a>) – contradiction.


category_id: SetxSet
property_id: well-copowered
reason:
Since it is an elementary topos and is locally essentially small, it is well-copowered (by <a href="/category-implication/topos_well-copowered_criterion">this result</a>).


category_id: SetxSet
property_id: zero morphisms
reason:
Assume for contradiction that it has zero morphisms. Then it would have coproducts and would have a generating set and would have zero morphisms, so it would have a generator (by <a href="/category-implication/generator_via_coproduct">this result</a>) – contradiction.


category_id: SetxSet
property_id: trivial
reason:
Assume for contradiction that it is trivial. Then it would be finitary algebraic (by <a href="/category-implication/trivial_consequence">this result</a>) – contradiction.


category_id: Set*
property_id: locally finitely multi-presentable
reason:
Since it has connected limits and is finitely accessible, it is locally finitely multi-presentable (by <a href="/category-implication/locally_finitely_multi-presentable_definition">this result</a>).


category_id: Set*
property_id: normal
reason:
Since it is pointed and has a subobject classifier, it is normal (by <a href="/category-implication/subobject_classifier_pointed_case">this result</a>).


category_id: Set*
property_id: counital
reason:
Since it is co-Malcev and is pointed, it is counital (by <a href="/category-implication/dual_malcev_implies_unital">this result</a>).


category_id: Set*
property_id: quotient object classifier
reason:
Assume for contradiction that it has a quotient object classifier. Then it would be pointed and would have a quotient object classifier, so it would be conormal (by <a href="/category-implication/dual_subobject_classifier_pointed_case">this result</a>) – contradiction.


category_id: Set*
property_id: regular quotient object classifier
reason:
Assume for contradiction that it has a regular quotient object classifier. Then it would be epi-regular and would have a regular quotient object classifier, so it would have a quotient object classifier (by <a href="/category-implication/dual_regular_subobjects_suffice">this result</a>) – contradiction.


category_id: Top*
property_id: cocomplete
reason:
Since it has coequalizers and has coproducts, it is cocomplete (by <a href="/category-implication/dual_complete_characterization">this result</a>).


category_id: Top*
property_id: locally ℵ₁-presentable
reason:
Assume for contradiction that it is locally ℵ₁-presentable. Then it would be locally presentable (by <a href="/category-implication/locally_countably_presentable_raise">this result</a>) – contradiction.


category_id: Top*
property_id: ℵ₁-accessible
reason:
Assume for contradiction that it is ℵ₁-accessible. Then it would be cocomplete and would be ℵ₁-accessible, so it would be locally ℵ₁-presentable (by <a href="/category-implication/locally_presentable_definition_countable">this result</a>) – contradiction.


category_id: Top*
property_id: accessible
reason:
Assume for contradiction that it is accessible. Then it would be accessible and would be cocomplete, so it would be locally presentable (by <a href="/category-implication/locally_presentable_definition">this result</a>) – contradiction.


category_id: Top*
property_id: locally multi-presentable
reason:
Assume for contradiction that it is locally multi-presentable. Then it would be accessible (by <a href="/category-implication/locally_multi-presentable_definition">this result</a>) – contradiction.


category_id: Top*
property_id: locally poly-presentable
reason:
Assume for contradiction that it is locally poly-presentable. Then it would be accessible (by <a href="/category-implication/locally_poly-presentable_definition">this result</a>) – contradiction.


category_id: Top*
property_id: self-dual
reason:
Assume for contradiction that it is self-dual. Then it would have cocartesian cofiltered limits and would be self-dual, so it would have cartesian filtered colimits (by <a href="/category-implication/self-dual_cocartesian cofiltered limits">this result</a>) – contradiction.


category_id: Pos
property_id: finite products
reason:
Since it is distributive, it has finite products (by <a href="/category-implication/distributive_assumption">this result</a>).


category_id: Pos
property_id: finitely complete
reason:
Since it has exact filtered colimits, it is finitely complete (by <a href="/category-implication/exact_filtered_colimits_condition">this result</a>).


category_id: Pos
property_id: powers
reason:
Since it is cartesian closed and has copowers, it has powers (by <a href="/category-implication/power_construction">this result</a>).


category_id: Pos
property_id: countable powers
reason:
Since it has powers, it has countable powers (by <a href="/category-implication/powers_include_countable_powers">this result</a>).


category_id: Pos
property_id: multi-algebraic
reason:
Assume for contradiction that it is multi-algebraic. Then it would be a generalized variety (by <a href="/category-implication/multi-algebraic_another_definition">this result</a>) – contradiction.


category_id: Pos
property_id: locally cartesian closed
reason:
Assume for contradiction that it is locally cartesian closed. Then it would have coequalizers and would be finitely complete and would be locally cartesian closed, so it would be regular (by <a href="/category-implication/lcc_implies_regular">this result</a>) – contradiction.


category_id: PMet
property_id: thin
reason:
Assume for contradiction that it is thin. Then it would have a terminal object and would be thin, so it would have powers (by <a href="/category-implication/thin_power">this result</a>) – contradiction.


category_id: Sch
property_id: binary products
reason:
Since it has pullbacks and has a terminal object, it has binary products (by <a href="/category-implication/binary_products_criterion">this result</a>).


category_id: Sch
property_id: equalizers
reason:
Since it has binary products and has pullbacks, it has equalizers (by <a href="/category-implication/equalizers_criterion">this result</a>).


category_id: Sch
property_id: essentially small
reason:
Assume for contradiction that it is essentially small. Then it would have a generating set (by <a href="/category-implication/essentially_small_consequences">this result</a>) – contradiction.


category_id: Sch
property_id: generator
reason:
Assume for contradiction that it has a generator. Then it would have a generating set (by <a href="/category-implication/generator_consequence">this result</a>) – contradiction.


category_id: Sch
property_id: thin
reason:
Assume for contradiction that it is thin. Then it would have a generating set (by <a href="/category-implication/thin_consequences">this result</a>) – contradiction.


category_id: Sch
property_id: accessible
reason:
Assume for contradiction that it is accessible. Then it would have a generating set (by <a href="/category-implication/accessible_trivial_consequence">this result</a>) – contradiction.


category_id: Sch
property_id: locally presentable
reason:
Assume for contradiction that it is locally presentable. Then it would be accessible (by <a href="/category-implication/locally_presentable_definition">this result</a>) – contradiction.


category_id: Sch
property_id: ℵ₁-accessible
reason:
Assume for contradiction that it is ℵ₁-accessible. Then it would be accessible (by <a href="/category-implication/countably_accessible_special_case">this result</a>) – contradiction.


category_id: Rel
property_id: zero morphisms
reason:
Since it has kernels, it has zero morphisms (by <a href="/category-implication/kernels_condition">this result</a>).


category_id: Rel
property_id: cokernels
reason:
Since it has kernels and is self-dual, it has cokernels (by <a href="/category-implication/self-dual_kernels">this result</a>).


category_id: Rel
property_id: coquotients of cocongruences
reason:
Since it has quotients of congruences and is self-dual, it has coquotients of cocongruences (by <a href="/category-implication/self-dual_quotients of congruences">this result</a>).


category_id: Rel
property_id: CSP
reason:
Since it satisfies CIP and is self-dual, it satisfies CSP (by <a href="/category-implication/self-dual_CIP">this result</a>).


category_id: Rel
property_id: well-copowered
reason:
Since it is self-dual and is well-powered, it is well-copowered (by <a href="/category-implication/self-dual_well-powered">this result</a>).


category_id: Rel
property_id: cogenerator
reason:
Since it has a generator and is self-dual, it has a cogenerator (by <a href="/category-implication/self-dual_generator">this result</a>).


category_id: Rel
property_id: equalizers
reason:
Assume for contradiction that it has equalizers. Then it would be Cauchy complete (by <a href="/category-implication/equalizers_consequence">this result</a>) – contradiction.


category_id: Rel
property_id: sequential colimits
reason:
Assume for contradiction that it has sequential colimits. Then it would be Cauchy complete (by <a href="/category-implication/sequential_colimits_consequence">this result</a>) – contradiction.


category_id: Rel
property_id: left cancellative
reason:
Assume for contradiction that it is left cancellative. Then it would be Cauchy complete (by <a href="/category-implication/cauchy_complete_criterion">this result</a>) – contradiction.


category_id: Rel
property_id: accessible
reason:
Assume for contradiction that it is accessible. Then it would be Cauchy complete (by <a href="/category-implication/accessible_require_Cauchy_complete">this result</a>) – contradiction.


category_id: Rel
property_id: sequential limits
reason:
Assume for contradiction that it has sequential limits. Then it would be Cauchy complete (by <a href="/category-implication/dual_sequential_colimits_consequence">this result</a>) – contradiction.


category_id: Rel
property_id: coaccessible
reason:
Assume for contradiction that it is coaccessible. Then it would be Cauchy complete (by <a href="/category-implication/dual_accessible_require_Cauchy_complete">this result</a>) – contradiction.


category_id: Rel
property_id: conormal
reason:
Assume for contradiction that it is conormal. Then it would be conormal and would be self-dual, so it would be normal (by <a href="/category-implication/self-dual_conormal">this result</a>) – contradiction.


category_id: Set_f
property_id: pullbacks
reason:
Since it is locally cartesian closed, it has pullbacks (by <a href="/category-implication/pullbacks_are_local_products">this result</a>).


category_id: Set_f
property_id: locally multi-presentable
reason:
Assume for contradiction that it is locally multi-presentable. Then it would be multi-cocomplete (by <a href="/category-implication/locally_multi-presentable_another_definition">this result</a>) – contradiction.


category_id: Set_f
property_id: connected limits
reason:
Assume for contradiction that it has connected limits. Then it would be accessible and would have connected limits, so it would be locally multi-presentable (by <a href="/category-implication/locally_multi-presentable_definition">this result</a>) – contradiction.


category_id: Set_f
property_id: wide pullbacks
reason:
Assume for contradiction that it has wide pullbacks. Then it would have equalizers and would have wide pullbacks, so it would have connected limits (by <a href="/category-implication/connected_limits_characterization">this result</a>) – contradiction.


category_id: Set_f
property_id: cofiltered limits
reason:
Assume for contradiction that it has cofiltered limits. Then it would have cofiltered limits and would have pullbacks, so it would have wide pullbacks (by <a href="/category-implication/wide_pullbacks_criterion">this result</a>) – contradiction.


category_id: Set_f
property_id: codistributive
reason:
Assume for contradiction that it is codistributive. Then it would have finite products (by <a href="/category-implication/dual_distributive_assumption">this result</a>) – contradiction.


category_id: Set_f
property_id: core-thin
reason:
Assume for contradiction that it is core-thin. Then it would have binary copowers and would be core-thin, so it would be thin (by <a href="/category-implication/dual_core-thin_products_thin">this result</a>) – contradiction.


category_id: Set_f
property_id: cocartesian coclosed
reason:
Assume for contradiction that it is cocartesian coclosed. Then it would be cocartesian coclosed and would have a strict initial object, so it would be thin (by <a href="/category-implication/dual_ccc_no_strict_terminal">this result</a>) – contradiction.


category_id: Sh(X)
property_id: finite products
reason:
Since it is infinitary distributive, it has finite products (by <a href="/category-implication/infinitary_distributive_assumption">this result</a>).


category_id: Sh(X)
property_id: well-powered
reason:
Since it is locally essentially small and has a subobject classifier, it is well-powered (by <a href="/category-implication/subobject_classifier_well-powered">this result</a>).


category_id: Sh(X)
property_id: generating set
reason:
Since it is a Grothendieck topos, it has a generating set (by <a href="/category-implication/grothendieck_topos_definition">this result</a>).


category_id: Sh(X)
property_id: exact filtered colimits
reason:
Since it is a Grothendieck topos, it has exact filtered colimits (by <a href="/category-implication/grothendieck_topos_consequence">this result</a>).


category_id: Sh(X)
property_id: locally presentable
reason:
Since it is a Grothendieck topos, it is locally presentable (by <a href="/category-implication/grothendieck_topos_consequence">this result</a>).


category_id: Man
property_id: equalizers
reason:
Assume for contradiction that it has equalizers. Then it would have binary products and would have equalizers, so it would have pullbacks (by <a href="/category-implication/pullbacks_criterion">this result</a>) – contradiction.


category_id: Man
property_id: essentially finite
reason:
Assume for contradiction that it is essentially finite. Then it would be essentially finite and would have finite powers, so it would be thin (by <a href="/category-implication/freyd_finite">this result</a>) – contradiction.


category_id: Man
property_id: coproducts
reason:
Assume for contradiction that it has coproducts. Then it would have copowers (by <a href="/category-implication/dual_products_consequences">this result</a>) – contradiction.


category_id: Z
property_id: essentially small
reason:
Assume for contradiction that it is essentially small. Then it would be locally essentially small (by <a href="/category-implication/essentially_small_consequences">this result</a>) – contradiction.


category_id: Z
property_id: locally small
reason:
Assume for contradiction that it is locally small. Then it would be locally essentially small (by <a href="/category-implication/locally_small_consequence">this result</a>) – contradiction.


category_id: Z
property_id: groupoid
reason:
Assume for contradiction that it is a groupoid. Then it would be well-powered (by <a href="/category-implication/groupoid_consequence">this result</a>) – contradiction.


category_id: Z
property_id: thin
reason:
Assume for contradiction that it is thin. Then it would be locally essentially small (by <a href="/category-implication/thin_consequences">this result</a>) – contradiction.


category_id: Z
property_id: preadditive
reason:
Assume for contradiction that it is preadditive. Then it would be locally essentially small (by <a href="/category-implication/preadditive_consequence">this result</a>) – contradiction.


category_id: Z
property_id: accessible
reason:
Assume for contradiction that it is accessible. Then it would be well-powered (by <a href="/category-implication/accessible_well-powered">this result</a>) – contradiction.


category_id: Z
property_id: coaccessible
reason:
Assume for contradiction that it is coaccessible. Then it would be locally essentially small (by <a href="/category-implication/dual_accessible_locally_small">this result</a>) – contradiction.


category_id: BG_f
property_id: essentially small
reason:
Since it is small, it is essentially small (by <a href="/category-implication/small_consequence">this result</a>).


category_id: BG_f
property_id: locally small
reason:
Since it is small, it is locally small (by <a href="/category-implication/small_consequence">this result</a>).


category_id: BG_f
property_id: countable
reason:
Since it is finite, it is countable (by <a href="/category-implication/finite_consequence">this result</a>).


category_id: BG_f
property_id: essentially finite
reason:
Since it is finite, it is essentially finite (by <a href="/category-implication/finite_consequence">this result</a>).


category_id: BG_f
property_id: essentially countable
reason:
Since it is essentially finite, it is essentially countable (by <a href="/category-implication/essentially_finite_raise">this result</a>).


category_id: BG_f
property_id: finitely accessible
reason:
Since it is Cauchy complete and is essentially finite, it is finitely accessible (by <a href="/category-implication/finite_filtered_colimits">this result</a>).


category_id: BG_f
property_id: wide pullbacks
reason:
Since it is essentially finite and has pullbacks, it has wide pullbacks (by <a href="/category-implication/wide_pullbacks_finite">this result</a>).


category_id: BG_f
property_id: strongly connected
reason:
Since it is connected and is a groupoid, it is strongly connected (by <a href="/category-implication/groupoid_connected">this result</a>).


category_id: BG_f
property_id: Grothendieck abelian
reason:
Assume for contradiction that it is Grothendieck abelian. Then it would be Grothendieck abelian and would be self-dual, so it would be trivial (by <a href="/category-implication/grothendieck_abelian_self-dual">this result</a>) – contradiction.


category_id: BG_f
property_id: essentially discrete
reason:
Assume for contradiction that it is essentially discrete. Then it would be connected and would be essentially discrete, so it would be trivial (by <a href="/category-implication/essentially_discrete_trivial">this result</a>) – contradiction.


category_id: BG_f
property_id: discrete
reason:
Assume for contradiction that it is discrete. Then it would be essentially discrete (by <a href="/category-implication/discrete_consequences">this result</a>) – contradiction.


category_id: BG_f
property_id: thin
reason:
Assume for contradiction that it is thin. Then it would be a groupoid and would be thin, so it would be essentially discrete (by <a href="/category-implication/essentially_discrete_characterization">this result</a>) – contradiction.


category_id: BG_f
property_id: cosifted
reason:
Assume for contradiction that it is cosifted. Then it would be cosifted and would be right cancellative, so it would be thin (by <a href="/category-implication/dual_sifted_left_cancellative_implies_thin">this result</a>) – contradiction.


category_id: BG_c
property_id: essentially countable
reason:
Since it is countable, it is essentially countable (by <a href="/category-implication/countable_consequence">this result</a>).


category_id: BN
property_id: core-thin
reason:
Since it is gaunt, it is core-thin (by <a href="/category-implication/gaunt_characterization">this result</a>).


category_id: BN
property_id: skeletal
reason:
Since it is gaunt, it is skeletal (by <a href="/category-implication/gaunt_characterization">this result</a>).


category_id: BN
property_id: pushouts
reason:
Since it has pullbacks and is self-dual, it has pushouts (by <a href="/category-implication/self-dual_pullbacks">this result</a>).


category_id: BN
property_id: right cancellative
reason:
Since it is left cancellative and is self-dual, it is right cancellative (by <a href="/category-implication/self-dual_left cancellative">this result</a>).


category_id: BN
property_id: locally cocartesian coclosed
reason:
Since it is locally cartesian closed and is self-dual, it is locally cocartesian coclosed (by <a href="/category-implication/self-dual_locally cartesian closed">this result</a>).


category_id: BN
property_id: balanced
reason:
Assume for contradiction that it is balanced. Then it would be balanced and would be left cancellative and would be right cancellative, so it would be a groupoid (by <a href="/category-implication/groupoid_criterion">this result</a>) – contradiction.


category_id: BN
property_id: binary powers
reason:
Assume for contradiction that it has binary powers. Then it would have binary powers and would be core-thin, so it would be thin (by <a href="/category-implication/core-thin_products_thin">this result</a>) – contradiction.


category_id: BN
property_id: wide pullbacks
reason:
Assume for contradiction that it has wide pullbacks. Then it would have cofiltered limits (by <a href="/category-implication/wide_pullbacks_criterion">this result</a>) – contradiction.


category_id: BN
property_id: filtered colimits
reason:
Assume for contradiction that it has filtered colimits. Then it would have filtered colimits and would be self-dual, so it would have cofiltered limits (by <a href="/category-implication/self-dual_filtered colimits">this result</a>) – contradiction.


category_id: BN
property_id: wide pushouts
reason:
Assume for contradiction that it has wide pushouts. Then it would have filtered colimits (by <a href="/category-implication/dual_wide_pullbacks_criterion">this result</a>) – contradiction.


category_id: BN
property_id: sequential colimits
reason:
Assume for contradiction that it has sequential colimits. Then it would be self-dual and would have sequential colimits, so it would have sequential limits (by <a href="/category-implication/self-dual_sequential colimits">this result</a>) – contradiction.


category_id: BOn
property_id: cofiltered
reason:
Since it has equalizers and is semi-strongly connected, it is cofiltered (by <a href="/category-implication/dual_filtered_via_equalizers">this result</a>).


category_id: BOn
property_id: pointed
reason:
Assume for contradiction that it is pointed. Then it would have an initial object (by <a href="/category-implication/pointed_characterization">this result</a>) – contradiction.


category_id: BOn
property_id: disjoint finite coproducts
reason:
Assume for contradiction that it has disjoint finite coproducts. Then it would have disjoint finite coproducts and would be locally cartesian closed, so it would be extensive (by <a href="/category-implication/lcc_implies_extensive">this result</a>) – contradiction.


category_id: BOn
property_id: multi-cocomplete
reason:
Assume for contradiction that it is multi-cocomplete. Then it would have a multi-initial object (by <a href="/category-implication/dual_multi-terminal_special_case">this result</a>) – contradiction.


category_id: 2
property_id: direct
reason:
Since it is discrete, it is direct (by <a href="/category-implication/discrete_consequences">this result</a>).


category_id: 2
property_id: one-way
reason:
Since it is direct, it is one-way (by <a href="/category-implication/direct_implies_one-way">this result</a>).


category_id: 2
property_id: skeletal
reason:
Since it is direct, it is skeletal (by <a href="/category-implication/direct_implies_one-way">this result</a>).


category_id: 2
property_id: core-thin
reason:
Since it is one-way, it is core-thin (by <a href="/category-implication/one-way_implies_core-thin">this result</a>).


category_id: 2
property_id: sequential limits
reason:
Since it is direct, it has sequential limits (by <a href="/category-implication/direct_implies_sequential_limits">this result</a>).


category_id: 2
property_id: reflexive coequalizers
reason:
Since it is one-way, it has reflexive coequalizers (by <a href="/category-implication/one-way_reflexive">this result</a>).


category_id: 2
property_id: gaunt
reason:
Since it is core-thin and is skeletal, it is gaunt (by <a href="/category-implication/gaunt_characterization">this result</a>).


category_id: 2
property_id: essentially discrete
reason:
Since it is discrete, it is essentially discrete (by <a href="/category-implication/discrete_consequences">this result</a>).


category_id: 2
property_id: groupoid
reason:
Since it is essentially discrete, it is a groupoid (by <a href="/category-implication/essentially_discrete_characterization">this result</a>).


category_id: 2
property_id: thin
reason:
Since it is core-thin and is a groupoid, it is thin (by <a href="/category-implication/thin_groupoids">this result</a>).


category_id: 2
property_id: connected limits
reason:
Since it is essentially discrete, it has connected limits (by <a href="/category-implication/essentially_discrete_consequence">this result</a>).


category_id: 2
property_id: generator
reason:
Since it is inhabited and is thin, it has a generator (by <a href="/category-implication/thin_inhabited_consequence">this result</a>).


category_id: 2
property_id: binary powers
reason:
Since it is thin, it has binary powers (by <a href="/category-implication/thin_binary_power">this result</a>).


category_id: 2
property_id: inverse
reason:
Since it is finite and is one-way and is skeletal, it is inverse (by <a href="/category-implication/dual_direct_criterion">this result</a>).


category_id: 2
property_id: connected colimits
reason:
Since it is essentially discrete, it has connected colimits (by <a href="/category-implication/dual_essentially_discrete_consequence">this result</a>).


category_id: 2
property_id: coequalizers
reason:
Since it has connected colimits, it has coequalizers (by <a href="/category-implication/dual_connected_limits_characterization">this result</a>).


category_id: 2
property_id: cogenerator
reason:
Since it is inhabited and is thin, it has a cogenerator (by <a href="/category-implication/dual_thin_inhabited_consequence">this result</a>).


category_id: 2
property_id: binary copowers
reason:
Since it is thin, it has binary copowers (by <a href="/category-implication/dual_thin_binary_power">this result</a>).


category_id: 2
property_id: multi-complete
reason:
Since it is multi-cocomplete and is self-dual, it is multi-complete (by <a href="/category-implication/self-dual_multi-cocomplete">this result</a>).


category_id: 2
property_id: zero morphisms
reason:
Assume for contradiction that it has zero morphisms. Then it would be inhabited and would be thin and would have zero morphisms, so it would be trivial (by <a href="/category-implication/thin_zero_trivial">this result</a>) – contradiction.


category_id: 2
property_id: binary products
reason:
Assume for contradiction that it has binary products. Then it would have binary products and would be inhabited, so it would be cosifted (by <a href="/category-implication/dual_coproducts_implies_sifted">this result</a>) – contradiction.


category_id: 0
property_id: thin
reason:
Since it is one-way and has zero morphisms, it is thin (by <a href="/category-implication/one-way_zero">this result</a>).


category_id: 0
property_id: equalizers
reason:
Since it has binary products and has coreflexive equalizers, it has equalizers (by <a href="/category-implication/equalizers_via_coreflexive_equalizers">this result</a>).


category_id: 0
property_id: normal
reason:
Since it is mono-regular and is preadditive, it is normal (by <a href="/category-implication/normal_criterion">this result</a>).


category_id: 0
property_id: conormal
reason:
Since it is epi-regular and is preadditive, it is conormal (by <a href="/category-implication/dual_normal_criterion">this result</a>).


category_id: 0
property_id: binary coproducts
reason:
Since it has binary products and is self-dual, it has binary coproducts (by <a href="/category-implication/self-dual_binary products">this result</a>).


category_id: 0
property_id: generator
reason:
Assume for contradiction that it has a generator. Then it would be inhabited (by <a href="/category-implication/generator_consequence">this result</a>) – contradiction.


category_id: 0
property_id: disjoint finite coproducts
reason:
Assume for contradiction that it has disjoint finite coproducts. Then it would have disjoint finite coproducts and would be thin, so it would be trivial (by <a href="/category-implication/disjoint_coproducts_thin_trivial">this result</a>) – contradiction.


category_id: 0
property_id: connected
reason:
Assume for contradiction that it is connected. Then it would be connected and would be essentially discrete, so it would be trivial (by <a href="/category-implication/essentially_discrete_trivial">this result</a>) – contradiction.


category_id: 0
property_id: initial object
reason:
Assume for contradiction that it has an initial object. Then it would have binary coproducts and would have an initial object, so it would have finite coproducts (by <a href="/category-implication/dual_finite_products_characterization">this result</a>) – contradiction.


category_id: 0
property_id: cogenerator
reason:
Assume for contradiction that it has a cogenerator. Then it would be inhabited (by <a href="/category-implication/dual_generator_consequence">this result</a>) – contradiction.


category_id: real_interval
property_id: filtered
reason:
Since it is finitely cocomplete, it is filtered (by <a href="/category-implication/filtered_criterion">this result</a>).


category_id: real_interval
property_id: thin
reason:
Since it has copowers and is essentially small, it is thin (by <a href="/category-implication/dual_freyd_small">this result</a>).


category_id: real_interval
property_id: one-way
reason:
Since it is thin, it is one-way (by <a href="/category-implication/thin_consequences">this result</a>).


category_id: real_interval
property_id: left cancellative
reason:
Since it is thin, it is left cancellative (by <a href="/category-implication/thin_consequence">this result</a>).


category_id: real_interval
property_id: locally cartesian closed
reason:
Since it is semi-strongly connected and is thin, it is locally cartesian closed (by <a href="/category-implication/sequential_implies_lcc">this result</a>).


category_id: real_interval
property_id: right cancellative
reason:
Since it is thin, it is right cancellative (by <a href="/category-implication/dual_thin_consequence">this result</a>).


category_id: real_interval
property_id: complete
reason:
Since it is cocomplete and is essentially small and is thin, it is complete (by <a href="/category-implication/dual_cocomplete_thin_criterion">this result</a>).


category_id: real_interval
property_id: regular
reason:
Since it is finitely complete and is thin, it is regular (by <a href="/category-implication/thin_implies_regular">this result</a>).


category_id: real_interval
property_id: regular subobject classifier
reason:
Since it is finitely complete and is right cancellative, it has a regular subobject classifier (by <a href="/category-implication/regular_subobjects_trivial">this result</a>).


category_id: real_interval
property_id: cartesian closed
reason:
Since it is locally cartesian closed and has a terminal object, it is cartesian closed (by <a href="/category-implication/locally_cartesian_closed_with_terminal_is_closed">this result</a>).


category_id: real_interval
property_id: codistributive
reason:
Since it is distributive and is thin, it is codistributive (by <a href="/category-implication/distributive_duality">this result</a>).


category_id: real_interval
property_id: countably distributive
reason:
Since it is cartesian closed and has countable coproducts, it is countably distributive (by <a href="/category-implication/countably_distributive_criterion">this result</a>).


category_id: real_interval
property_id: exact filtered colimits
reason:
Since it has cartesian filtered colimits and is thin, it has exact filtered colimits (by <a href="/category-implication/thin_exact_filtered_colimits">this result</a>).


category_id: real_interval
property_id: Malcev
reason:
Since it is finitely complete and is thin, it is Malcev (by <a href="/category-implication/malcev_thin_condition">this result</a>).


category_id: real_interval
property_id: strict terminal object
reason:
Since it is right cancellative and has a terminal object, it has a strict terminal object (by <a href="/category-implication/dual_strict_initial_left_criterion">this result</a>).


category_id: real_interval
property_id: locally cocartesian coclosed
reason:
Since it is semi-strongly connected and is thin, it is locally cocartesian coclosed (by <a href="/category-implication/dual_sequential_implies_lcc">this result</a>).


category_id: real_interval
property_id: coregular
reason:
Since it is finitely cocomplete and is thin, it is coregular (by <a href="/category-implication/dual_thin_implies_regular">this result</a>).


category_id: real_interval
property_id: locally copresentable
reason:
Since it is coaccessible and is complete, it is locally copresentable (by <a href="/category-implication/dual_locally_presentable_definition">this result</a>).


category_id: real_interval
property_id: regular quotient object classifier
reason:
Since it is finitely cocomplete and is left cancellative, it has a regular quotient object classifier (by <a href="/category-implication/dual_regular_subobjects_trivial">this result</a>).


category_id: real_interval
property_id: cocartesian coclosed
reason:
Since it has an initial object and is locally cocartesian coclosed, it is cocartesian coclosed (by <a href="/category-implication/dual_locally_cartesian_closed_with_terminal_is_closed">this result</a>).


category_id: real_interval
property_id: countably codistributive
reason:
Since it is cocartesian coclosed and has countable products, it is countably codistributive (by <a href="/category-implication/dual_countably_distributive_criterion">this result</a>).


category_id: real_interval
property_id: infinitary codistributive
reason:
Since it is cocartesian coclosed and has products, it is infinitary codistributive (by <a href="/category-implication/dual_infinitary_distributive_criterion">this result</a>).


category_id: real_interval
property_id: cocartesian cofiltered limits
reason:
Since it is cocartesian coclosed and has cofiltered limits, it has cocartesian cofiltered limits (by <a href="/category-implication/dual_ccc_cartesian_filtered_colimits">this result</a>).


category_id: real_interval
property_id: exact cofiltered limits
reason:
Since it has cocartesian cofiltered limits and is thin, it has exact cofiltered limits (by <a href="/category-implication/dual_thin_exact_filtered_colimits">this result</a>).


category_id: real_interval
property_id: co-Malcev
reason:
Since it is finitely cocomplete and is thin, it is co-Malcev (by <a href="/category-implication/dual_malcev_thin_condition">this result</a>).


category_id: real_interval
property_id: groupoid
reason:
Assume for contradiction that it is a groupoid. Then it would be a groupoid and would be thin, so it would be essentially discrete (by <a href="/category-implication/essentially_discrete_characterization">this result</a>) – contradiction.


category_id: real_interval
property_id: strongly connected
reason:
Assume for contradiction that it is strongly connected. Then it would be strongly connected and would be thin, so it would be trivial (by <a href="/category-implication/thin_not_strongly_connected">this result</a>) – contradiction.


category_id: real_interval
property_id: disjoint finite products
reason:
Assume for contradiction that it has disjoint finite products. Then it would have disjoint finite products and would be thin, so it would be trivial (by <a href="/category-implication/dual_disjoint_coproducts_thin_trivial">this result</a>) – contradiction.


category_id: N_oo
property_id: countable products
reason:
Since it has finite products and has sequential limits, it has countable products (by <a href="/category-implication/countable_products_criterion">this result</a>).


category_id: N_oo
property_id: thin
reason:
Since it has binary powers and is core-thin, it is thin (by <a href="/category-implication/core-thin_products_thin">this result</a>).


category_id: N_oo
property_id: products
reason:
Since it has countable products and is essentially countable and is thin, it has products (by <a href="/category-implication/thin_countable_product_reduction">this result</a>).


category_id: N_oo
property_id: cocomplete
reason:
Since it is complete and is essentially small and is thin, it is cocomplete (by <a href="/category-implication/cocomplete_thin_criterion">this result</a>).


category_id: N_oo
property_id: strict terminal object
reason:
Since it is left cancellative and has a terminal object, it has a strict terminal object (by <a href="/category-implication/dual_strict_initial_right_criterion">this result</a>).


category_id: N_oo
property_id: groupoid
reason:
Assume for contradiction that it is a groupoid. Then it would be self-dual (by <a href="/category-implication/groupoid_consequence">this result</a>) – contradiction.


category_id: N
property_id: ℵ₁-accessible
reason:
Since it is essentially countable and is thin, it is ℵ₁-accessible (by <a href="/category-implication/countably_accessible_thin">this result</a>).


category_id: N
property_id: finite coproducts
reason:
Since it is finitely cocomplete, it has finite coproducts (by <a href="/category-implication/dual_finitely_complete_characterization">this result</a>).


category_id: N
property_id: copowers
reason:
Since it has an initial object and is thin, it has copowers (by <a href="/category-implication/dual_thin_power">this result</a>).


category_id: N
property_id: countable copowers
reason:
Since it has copowers, it has countable copowers (by <a href="/category-implication/dual_powers_include_countable_powers">this result</a>).


category_id: N
property_id: countably distributive
reason:
Assume for contradiction that it is countably distributive. Then it would have countable coproducts (by <a href="/category-implication/countably_distributive_assumption">this result</a>) – contradiction.


category_id: N
property_id: complete
reason:
Assume for contradiction that it is complete. Then it would be complete and would be essentially small and would be thin, so it would be cocomplete (by <a href="/category-implication/cocomplete_thin_criterion">this result</a>) – contradiction.


category_id: N
property_id: products
reason:
Assume for contradiction that it has products. Then it would have equalizers and would have products, so it would be complete (by <a href="/category-implication/complete_characterization">this result</a>) – contradiction.


category_id: N
property_id: finite products
reason:
Assume for contradiction that it has finite products. Then it would have cofiltered limits and would have finite products, so it would have products (by <a href="/category-implication/products_criterion">this result</a>) – contradiction.


category_id: N
property_id: terminal object
reason:
Assume for contradiction that it has a terminal object. Then it would have binary products and would have a terminal object, so it would have finite products (by <a href="/category-implication/finite_products_characterization">this result</a>) – contradiction.


category_id: N
property_id: self-dual
reason:
Assume for contradiction that it is self-dual. Then it would be finitely cocomplete and would be self-dual, so it would be finitely complete (by <a href="/category-implication/self-dual_finitely cocomplete">this result</a>) – contradiction.


category_id: On
property_id: generating set
reason:
Since it is thin, it has a generating set (by <a href="/category-implication/thin_consequences">this result</a>).


category_id: On
property_id: cogenerating set
reason:
Since it is thin, it has a cogenerating set (by <a href="/category-implication/dual_thin_consequences">this result</a>).


category_id: On
property_id: essentially small
reason:
Assume for contradiction that it is essentially small. Then it would be well-copowered (by <a href="/category-implication/essentially_small_consequences">this result</a>) – contradiction.


category_id: On
property_id: preadditive
reason:
Assume for contradiction that it is preadditive. Then it would have finite coproducts and would be preadditive, so it would have finite products (by <a href="/category-implication/preadditive_products_criterion">this result</a>) – contradiction.


category_id: On
property_id: finitary algebraic
reason:
Assume for contradiction that it is finitary algebraic. Then it would be well-copowered (by <a href="/category-implication/algebraic_well-copowered">this result</a>) – contradiction.


category_id: On
property_id: accessible
reason:
Assume for contradiction that it is accessible. Then it would be accessible and would have pushouts, so it would be well-copowered (by <a href="/category-implication/accessible_well-copowered">this result</a>) – contradiction.


category_id: On
property_id: coaccessible
reason:
Assume for contradiction that it is coaccessible. Then it would be well-copowered (by <a href="/category-implication/dual_accessible_well-powered">this result</a>) – contradiction.


category_id: On
property_id: quotient object classifier
reason:
Assume for contradiction that it has a quotient object classifier. Then it would be locally essentially small and would have a quotient object classifier, so it would be well-copowered (by <a href="/category-implication/dual_subobject_classifier_well-powered">this result</a>) – contradiction.


category_id: On
property_id: self-dual
reason:
Assume for contradiction that it is self-dual. Then it would be cocomplete and would be self-dual, so it would be complete (by <a href="/category-implication/self-dual_cocomplete">this result</a>) – contradiction.


category_id: Z_div
property_id: thin
reason:
Since it is essentially small and has powers, it is thin (by <a href="/category-implication/freyd_small">this result</a>).


category_id: Z_div
property_id: equalizers
reason:
Since it is thin, it has equalizers (by <a href="/category-implication/thin_consequence">this result</a>).


category_id: Z_div
property_id: natural numbers object
reason:
Since it has finite products and is thin, it has a natural numbers object (by <a href="/category-implication/nno_thin">this result</a>).


category_id: Z_div
property_id: coproducts
reason:
Since it has filtered colimits and has finite coproducts, it has coproducts (by <a href="/category-implication/dual_products_criterion">this result</a>).


category_id: Z_div
property_id: cartesian closed
reason:
Assume for contradiction that it is cartesian closed. Then it would be cartesian closed and would have countable coproducts, so it would be countably distributive (by <a href="/category-implication/countably_distributive_criterion">this result</a>) – contradiction.


category_id: Z_div
property_id: cartesian filtered colimits
reason:
Assume for contradiction that it has cartesian filtered colimits. Then it would have cartesian filtered colimits and would have coproducts and would be distributive, so it would be infinitary distributive (by <a href="/category-implication/infinite_distributive_filtered_criterion">this result</a>) – contradiction.


category_id: Z_div
property_id: semi-strongly connected
reason:
Assume for contradiction that it is semi-strongly connected. Then it would be semi-strongly connected and would be thin, so it would be locally cartesian closed (by <a href="/category-implication/sequential_implies_lcc">this result</a>) – contradiction.


category_id: Z_div
property_id: exact cofiltered limits
reason:
Assume for contradiction that it has exact cofiltered limits. Then it would have cocartesian cofiltered limits (by <a href="/category-implication/dual_exact_includes_cartesian_filtered_colimits">this result</a>) – contradiction.


category_id: Delta
property_id: Cauchy complete
reason:
Since it has coequalizers, it is Cauchy complete (by <a href="/category-implication/dual_equalizers_consequence">this result</a>).


category_id: Delta
property_id: coquotients of cocongruences
reason:
Since it is core-thin, it has coquotients of cocongruences (by <a href="/category-implication/dual_core-hin_quotients">this result</a>).


category_id: Delta
property_id: equalizers
reason:
Assume for contradiction that it has equalizers. Then it would have coreflexive equalizers (by <a href="/category-implication/coreflexive_equalizers_are_equalizers">this result</a>) – contradiction.


category_id: Delta
property_id: direct
reason:
Assume for contradiction that it is direct. Then it would have sequential limits (by <a href="/category-implication/direct_implies_sequential_limits">this result</a>) – contradiction.


category_id: Delta
property_id: binary coproducts
reason:
Assume for contradiction that it has binary coproducts. Then it would have binary coproducts and would have coequalizers, so it would have pushouts (by <a href="/category-implication/dual_pullbacks_criterion">this result</a>) – contradiction.


category_id: Delta
property_id: inverse
reason:
Assume for contradiction that it is inverse. Then it would have sequential colimits (by <a href="/category-implication/dual_direct_implies_sequential_limits">this result</a>) – contradiction.


category_id: Delta
property_id: one-way
reason:
Assume for contradiction that it is one-way. Then it would have coreflexive equalizers (by <a href="/category-implication/dual_one-way_reflexive">this result</a>) – contradiction.


category_id: Delta
property_id: initial object
reason:
Assume for contradiction that it has an initial object. Then it would be cofiltered (by <a href="/category-implication/dual_terminal_object_yields_filtered">this result</a>) – contradiction.


category_id: Delta
property_id: self-dual
reason:
Assume for contradiction that it is self-dual. Then it would be self-dual and would have a terminal object, so it would have an initial object (by <a href="/category-implication/self-dual_terminal object">this result</a>) – contradiction.


category_id: 1
property_id: finitary algebraic
reason:
Since it is trivial, it is finitary algebraic (by <a href="/category-implication/trivial_consequence">this result</a>).


category_id: 1
property_id: products
reason:
Since it has cofiltered limits and has finite products, it has products (by <a href="/category-implication/products_criterion">this result</a>).


category_id: 1
property_id: Grothendieck topos
reason:
Since it is trivial, it is a Grothendieck topos (by <a href="/category-implication/trivial_consequence">this result</a>).


category_id: 1
property_id: split abelian
reason:
Since it is trivial, it is split abelian (by <a href="/category-implication/trivial_consequence">this result</a>).


category_id: 1
property_id: cokernels
reason:
Since it is preadditive and has quotients of congruences and is regular, it has cokernels (by <a href="/category-implication/cokernels_via_congruence_quotients">this result</a>).


category_id: 1
property_id: zero morphisms
reason:
Since it is normal, it has zero morphisms (by <a href="/category-implication/normal_condition">this result</a>).


category_id: 1
property_id: pointed
reason:
Since it has an initial object and has zero morphisms, it is pointed (by <a href="/category-implication/pointed_characterization">this result</a>).


category_id: 1
property_id: infinitary codistributive
reason:
Since it has cocartesian cofiltered limits and is codistributive and has products, it is infinitary codistributive (by <a href="/category-implication/dual_infinite_distributive_filtered_criterion">this result</a>).


category_id: 1
property_id: countably codistributive
reason:
Since it is infinitary codistributive, it is countably codistributive (by <a href="/category-implication/dual_infinitary_distributive_trivial">this result</a>).


category_id: 1
property_id: CSP
reason:
Since it has biproducts and has cofiltered limits and has cofiltered-limit-stable epimorphisms and has coproducts, it satisfies CSP (by <a href="/category-implication/dual_CIP_criterion">this result</a>).


category_id: 1
property_id: cocartesian coclosed
reason:
Since it is cocomplete and is essentially small and is infinitary codistributive and is thin, it is cocartesian coclosed (by <a href="/category-implication/dual_cartesian_closed_thin_criterion">this result</a>).


category_id: 1
property_id: quotient object classifier
reason:
Since it is epi-regular and has a regular quotient object classifier, it has a quotient object classifier (by <a href="/category-implication/dual_regular_subobjects_suffice">this result</a>).


category_id: 1
property_id: coextensive
reason:
Since it has disjoint finite products and is locally cocartesian coclosed, it is coextensive (by <a href="/category-implication/dual_lcc_implies_extensive">this result</a>).


category_id: 1
property_id: infinitary coextensive
reason:
Since it is coextensive and is complete and is locally cocartesian coclosed, it is infinitary coextensive (by <a href="/category-implication/dual_lcc_extensive_yields_infinitary">this result</a>).


category_id: walking_commutative_square
property_id: coproducts
reason:
Since it is infinitary distributive, it has coproducts (by <a href="/category-implication/infinitary_distributive_assumption">this result</a>).


category_id: walking_commutative_square
property_id: direct
reason:
Since it is finite and is one-way and is skeletal, it is direct (by <a href="/category-implication/direct_criterion">this result</a>).


category_id: walking_commutative_square
property_id: cartesian closed
reason:
Since it is complete and is essentially small and is infinitary distributive and is thin, it is cartesian closed (by <a href="/category-implication/cartesian_closed_thin_criterion">this result</a>).


category_id: walking_commutative_square
property_id: infinitary codistributive
reason:
Since it is infinitary distributive and is self-dual, it is infinitary codistributive (by <a href="/category-implication/self-dual_infinitary distributive">this result</a>).


category_id: walking_fork
property_id: thin
reason:
Assume for contradiction that it is thin. Then it would have binary powers (by <a href="/category-implication/thin_binary_power">this result</a>) – contradiction.


category_id: walking_idempotent
property_id: coreflexive equalizers
reason:
Since it has reflexive coequalizers and is self-dual, it has coreflexive equalizers (by <a href="/category-implication/self-dual_reflexive coequalizers">this result</a>).


category_id: walking_idempotent
property_id: epi-regular
reason:
Since it is mono-regular and is self-dual, it is epi-regular (by <a href="/category-implication/self-dual_mono-regular">this result</a>).


category_id: walking_idempotent
property_id: cofiltered
reason:
Since it is filtered and is self-dual, it is cofiltered (by <a href="/category-implication/self-dual_filtered">this result</a>).


category_id: walking_idempotent
property_id: kernels
reason:
Assume for contradiction that it has kernels. Then it would have kernels and would be preadditive, so it would have equalizers (by <a href="/category-implication/equalizers_via_kernels">this result</a>) – contradiction.


category_id: walking_idempotent
property_id: finite coproducts
reason:
Assume for contradiction that it has finite coproducts. Then it would have finite coproducts and would be preadditive, so it would have finite products (by <a href="/category-implication/preadditive_products_criterion">this result</a>) – contradiction.


category_id: walking_idempotent
property_id: coregular
reason:
Assume for contradiction that it is coregular. Then it would have coquotients of cocongruences and would be coregular and would be preadditive, so it would have kernels (by <a href="/category-implication/dual_cokernels_via_congruence_quotients">this result</a>) – contradiction.


category_id: walking_idempotent
property_id: coequalizers
reason:
Assume for contradiction that it has coequalizers. Then it would be Cauchy complete (by <a href="/category-implication/dual_equalizers_consequence">this result</a>) – contradiction.


category_id: walking_idempotent
property_id: binary coproducts
reason:
Assume for contradiction that it has binary coproducts. Then it would have binary coproducts and would have reflexive coequalizers, so it would have coequalizers (by <a href="/category-implication/dual_equalizers_via_coreflexive_equalizers">this result</a>) – contradiction.


category_id: walking_idempotent
property_id: pullbacks
reason:
Assume for contradiction that it has pullbacks. Then it would be essentially finite and would have pullbacks, so it would have wide pullbacks (by <a href="/category-implication/wide_pullbacks_finite">this result</a>) – contradiction.


category_id: walking_idempotent
property_id: pushouts
reason:
Assume for contradiction that it has pushouts. Then it would be essentially finite and would have pushouts, so it would have wide pushouts (by <a href="/category-implication/dual_wide_pullbacks_finite">this result</a>) – contradiction.


category_id: walking_idempotent
property_id: pointed
reason:
Assume for contradiction that it is pointed. Then it would have a terminal object (by <a href="/category-implication/dual_pointed_characterization">this result</a>) – contradiction.


category_id: walking_idempotent
property_id: initial object
reason:
Assume for contradiction that it has an initial object. Then it would have an initial object and would have zero morphisms, so it would be pointed (by <a href="/category-implication/pointed_characterization">this result</a>) – contradiction.


category_id: walking_idempotent
property_id: right cancellative
reason:
Assume for contradiction that it is right cancellative. Then it would be right cancellative and would have zero morphisms, so it would be thin (by <a href="/category-implication/dual_zero-no-mono">this result</a>) – contradiction.


category_id: walking_idempotent
property_id: binary copowers
reason:
Assume for contradiction that it has binary copowers. Then it would have binary copowers and would be core-thin, so it would be thin (by <a href="/category-implication/dual_core-thin_products_thin">this result</a>) – contradiction.


category_id: walking_isomorphism
property_id: essentially discrete
reason:
Since it is trivial, it is essentially discrete (by <a href="/category-implication/trivial_consequence">this result</a>).


category_id: walking_isomorphism
property_id: thin
reason:
Since it is essentially discrete, it is thin (by <a href="/category-implication/essentially_discrete_characterization">this result</a>).


category_id: walking_pair
property_id: cofiltered limits
reason:
Since it has filtered colimits and is self-dual, it has cofiltered limits (by <a href="/category-implication/self-dual_filtered colimits">this result</a>).


category_id: walking_pair
property_id: cosifted limits
reason:
Since it is self-dual and has sifted colimits, it has cosifted limits (by <a href="/category-implication/self-dual_sifted colimits">this result</a>).


category_id: walking_pair
property_id: groupoid
reason:
Assume for contradiction that it is a groupoid. Then it would have pullbacks (by <a href="/category-implication/groupoid_consequence">this result</a>) – contradiction.


category_id: walking_pair
property_id: terminal object
reason:
Assume for contradiction that it has a terminal object. Then it would be filtered (by <a href="/category-implication/terminal_object_yields_filtered">this result</a>) – contradiction.


category_id: walking_pair
property_id: pushouts
reason:
Assume for contradiction that it has pushouts. Then it would have pushouts and would be self-dual, so it would have pullbacks (by <a href="/category-implication/self-dual_pushouts">this result</a>) – contradiction.


category_id: walking_span
property_id: connected
reason:
Since it has an initial object, it is connected (by <a href="/category-implication/dual_multi-terminal_with_connected">this result</a>).


category_id: walking_span
property_id: coequalizers
reason:
Since it is thin, it has coequalizers (by <a href="/category-implication/dual_thin_consequence">this result</a>).


category_id: walking_span
property_id: finite copowers
reason:
Since it has countable copowers, it has finite copowers (by <a href="/category-implication/dual_countable_powers_include_finite_powers">this result</a>).


category_id: walking_span
property_id: binary copowers
reason:
Since it has finite copowers, it has binary copowers (by <a href="/category-implication/dual_finite_powers_consequences">this result</a>).


category_id: walking_splitting
property_id: terminal object
reason:
Since it is pointed, it has a terminal object (by <a href="/category-implication/dual_pointed_characterization">this result</a>).


category_id: walking_splitting
property_id: coequalizers
reason:
Since it has equalizers and is self-dual, it has coequalizers (by <a href="/category-implication/self-dual_equalizers">this result</a>).


category_id: walking_splitting
property_id: cofiltered-limit-stable epimorphisms
reason:
Since it has filtered-colimit-stable monomorphisms and is self-dual, it has cofiltered-limit-stable epimorphisms (by <a href="/category-implication/self-dual_filtered-colimit-stable monomorphisms">this result</a>).


category_id: walking_splitting
property_id: left cancellative
reason:
Assume for contradiction that it is left cancellative. Then it would be left cancellative and would have zero morphisms, so it would be thin (by <a href="/category-implication/zero-no-mono">this result</a>) – contradiction.


category_id: walking_splitting
property_id: binary coproducts
reason:
Assume for contradiction that it has binary coproducts. Then it would have binary coproducts and would have an initial object, so it would have finite coproducts (by <a href="/category-implication/dual_finite_products_characterization">this result</a>) – contradiction.


category_id: walking_splitting
property_id: multi-complete
reason:
Assume for contradiction that it is multi-complete. Then it would be multi-complete and would be self-dual, so it would be multi-cocomplete (by <a href="/category-implication/self-dual_multi-complete">this result</a>) – contradiction.


# functor_property_assignments

functor_id: abelianization
property_id: product-preserving
reason:
If $G$ is a group, the canonical homomorphism $(G^{\mathbb{N}})^{\mathrm{ab}} \to (G^{\mathrm{ab}})^{\mathbb{N}}$ is surjective, but does not need to be an isomorphism: otherwise, the inclusion $[G^{\mathbb{N}}, G^{\mathbb{N}}] \subseteq [G,G]^{\mathbb{N}}$ would be an equality. But this requires the commutator width of $G$ to be finite, which fails for $G = F_2$ for instance. See also <a href="https://arxiv.org/abs/1608.02220">The abelianization of inverse limits of groups</a>, Remark 0.0.7.


functor_id: abelianization
property_id: faithful
reason:
Both the inclusion $A_3 \hookrightarrow S_3$ and the trivial homomorphism $A_3 \to S_3$ are mapped to the trivial homomorphism $A_3 \to 1$.


functor_id: abelianization
property_id: conservative
reason:
The proper inclusion $S_3 \hookrightarrow S_4$ gets mapped to the trivial homomorphism $1 \to 1$, which is an isomorphism.


functor_id: forget_vector
property_id: finitary
reason:
For every algebraic category the forgetful functor to the category of sets preserves filtered colimits.


functor_id: forget_vector
property_id: epimorphism-preserving
reason:
This follows from the classifications of epimorphisms in the <a href="/category/Vect">category of vector spaces</a> and in the <a href="/category/Set">category of sets</a>.


functor_id: forget_vector
property_id: coequalizer-preserving
reason:
The coequalizer of $0,i_1 : K \to K^2$ in $\mathbf{Vect}$ is $p_2 : K^2 \to K$, but the coequalizer in $\mathbf{Set}$ is $(K \times K^*) \cup \{(0,0)\}$.


functor_id: free_group
property_id: monomorphism-preserving
reason:
This can be deduced from the description of the elements of a free group, but here is an abstract argument: Split monomorphisms are preserved by any functor. The only injective maps in $\mathbf{Set}$ that are not split are $\varnothing \hookrightarrow X$ (for non-empty $X$), and $F(\varnothing) \to F(X)$ is injective since $F(\varnothing)$ is the trivial group.


functor_id: free_group
property_id: faithful
reason:
A left adjoint is faithful if and only if its unit consists of monomorphisms. So we only need to check that every set <i>embeds</i> into (the underlying set of) its free group. But this is clear since $X$ already embeds into the free abelian group $\mathbb{Z}^{\oplus X}$, which is a quotient of the free group.


functor_id: free_group
property_id: conservative
reason:
Let $f : X \to Y$ be a map of sets such that $F(f) : F(X) \to F(Y)$ is an isomorphism of groups. We know that $F$ is faithful, so that it reflects monomorphisms. Thus, $f$ is injective. Choose a complement $U \subseteq Y$ of $f(X) \subseteq Y$. Then $F(X) \to F(Y) = F(X) \sqcup F(U)$ is an isomorphism. This implies $F(U)=1$ and hence $U = \varnothing$.


functor_id: free_group
property_id: full
reason:
The map $1 = \mathrm{Hom}(1,1) \to \mathrm{Hom}(F(1),F(1)) = \mathrm{Hom}(\mathbb{Z},\mathbb{Z}) = \mathbb{Z}$ is not surjective.


functor_id: free_group
property_id: equalizer-preserving
reason:
Let $f,g : \{a,b\} \rightrightarrows \{c,d\}$ be the two constant maps, $f \equiv c$, $g \equiv d$. Their equalizer is empty, and the free group on that equalizer is trivial. Now consider the induced group homomorphisms $F(f), F(g) : F(\{a,b\}) \rightrightarrows F(\{c,d\})$ and observe that $a \cdot b^{-1}$ lies in the kernel of both homomorphisms, hence in their equalizer.


functor_id: power_set_contravariant
property_id: representable
reason:
This is because there are natural bijections $P(X) \cong \mathrm{Hom}(X,2)$, sending a subset to its characteristic function.


functor_id: power_set_contravariant
property_id: monadic
reason:
See <a href="https://ncatlab.org/nlab/show/Sketches+of+an+Elephant" target="_blank">Johnstone</a>, Theorem 2.2.7.


functor_id: power_set_contravariant
property_id: epimorphism-preserving
reason:
If $f : X \to Y$ is injective, then $f^* : P(Y) \to P(X)$ is surjective, since for all $A \subseteq X$ we have $A = f^*(f_*(A))$.


functor_id: power_set_contravariant
property_id: coequalizer-preserving
reason:
The power set functor preserves reflexive coequalizers, but not all coequalizers, i.e. there are maps $f,g : Y \to X$ with equalizer $E \subseteq Y$ such that $P(Y) \twoheadrightarrow P(E)$ is not the coequalizer of $f^*,g^* : P(X) \to P(Y)$: Let $X=\{0,1\}$ and $Y=\{a,b,c\}$. Define maps $f,g:Y\to X$ by $f(a)=0$, $f(b)=0$, $f(c)=1$ and $g(a)=0$, $g(b)=1$, $g(c)=0$. Their equalizer is $E = \{a\}$, so that $P(E)$ has $2$ elements. For $S=\{0\}$ one has $f^*(S)=\{a,b\}$, $g^*(S)=\{a,c\}$, and for $S=\{1\}$ one has $f^*(S)=\{c\}$, $g^*(S)=\{b\}$. Thus the coequalizer of $f^*,g^*$ is obtained from $P(Y)$ by imposing the two relations $\{a,b\} \sim \{a,c\}$ and $\{c\} \sim \{b\}$. But then it as $8-2 = 6$ elements.


functor_id: power_set_contravariant
property_id: initial-object-preserving
reason:
In fact, the initial object does not even lie in the essential image since $P(X) \neq 0$ for all $X$.


functor_id: power_set_contravariant
property_id: finitary
reason:
Consider the sequence of projections $\cdots \twoheadrightarrow \{0,1\}^2 \twoheadrightarrow \{0,1\}^1$. Its limit in $\mathbf{Set}$ (i.e. colimit in $\mathbf{Set}^{\mathrm{op}}$) is $\{0,1\}^{\mathbb{N}}$, which is uncountable, so that $P(\{0,1\}^{\mathbb{N}})$ is also uncountable. But the colimit of the induced diagram $P(\{0,1\}^1) \hookrightarrow P(\{0,1\}^2) \hookrightarrow \cdots$ is countable since each $P(\{0,1\}^n)$ is finite.


functor_id: power_set_contravariant
property_id: full
reason:
The maps $f^* : P(Y) \to P(X)$ preserve the empty set, so take the constant map $P(X) \to P(X)$, $T \mapsto X$ for instance (where $X$ is non-empty).


functor_id: power_set_covariant
property_id: monomorphism-preserving
reason:
If $f : X \to Y$ is injective, then $f^* \circ f_* = \mathrm{id}_{P(X)}$, so that $f_*$ is injective.


functor_id: power_set_covariant
property_id: epimorphism-preserving
reason:
If $f : X \to Y$ is surjective, then $f_* \circ f^* = \mathrm{id}_{P(Y)}$, so that $f^*$ is surjective.


functor_id: power_set_covariant
property_id: faithful
reason:
Let $f,g : X \rightrightarrows Y$ be two maps with $f_* = g_* : P(X) \rightrightarrows P(Y)$. Then $\{f(x)\} = f_*(\{x\}) = g_*(\{x\}) = \{g(x)\}$ and hence $f(x) = g(x)$ for all $x \in X$.


functor_id: power_set_covariant
property_id: conservative
reason:
Assume that $f : X \to Y$ is a map such that $f_* : P(X) \to P(Y)$ is an isomorphism. There is some $A \subseteq X$ with $Y = f_*(A)$, this proves that $f$ is surjective. It is also injective: If $x,y \in X$ satisfy $f(x) = f(y)$, then $f_*(\{x\}) = f_*(\{y\})$, and hence $\{x\} = \{y\}$, i.e. $x = y$.


functor_id: power_set_covariant
property_id: full
reason:
Take any map $P(X) \to P(X)$ that does not preserve the empty set, say the constant map with value $X$ (for $X \neq \varnothing$).


functor_id: power_set_covariant
property_id: finitary
reason:
The filtered colimit $\mathbb{N} = \bigcup_{n \geq 0} \mathbb{N}_{\leq n}$ is not preserved by $P$, since $\bigcup_{n \geq 0} P(\mathbb{N}_{\leq n})$ just consists of the <i>finite</i> subsets of $\mathbb{N}$.


functor_id: power_set_covariant
property_id: equalizer-preserving
reason:
Any pair of distinct surjective maps $f,g : X \rightrightarrows Y$ provides a counterexample: Their equalizer $E$ is a proper subset of $X$, so that $P(E)$ cannot contain $X$. But $f_*(X) = Y = g_*(X)$ shows that $X$ is contained in the equalizer of $f_*,g_* : P(X) \rightrightarrows P(Y)$.


functor_id: power_set_covariant
property_id: coequalizer-preserving
reason:
Let $X := \{x,y\}$. Consider the two maps $x,y : \{0\} \rightrightarrows X$. Their coequalizer $Q = X / (x = y)$ has just one element, so that $P(Q)$ has two elements. The induced maps $x_*,y_* : P(\{0\}) \rightrightarrows P(X)$ (which already agree on the empty set) have coequalizer $P(X) / (\{x\} = \{y\})$, which has $3$ elements. So it cannot be $P(Q)$.


functor_id: abelianization
property_id: terminal-object-preserving
reason:
Since it is finite-product-preserving, it is terminal-object-preserving (by <a href="/functor-implication/finite_products_consequences">this result</a>).


functor_id: abelianization
property_id: cocontinuous
reason:
Since it is a left adjoint, it is cocontinuous (by <a href="/functor-implication/dual_right_adjoint_consequences">this result</a>).


functor_id: abelianization
property_id: coequalizer-preserving
reason:
Since it is cocontinuous, it is coequalizer-preserving (by <a href="/functor-implication/dual_continuous_consequences">this result</a>).


functor_id: abelianization
property_id: coproduct-preserving
reason:
Since it is cocontinuous, it is coproduct-preserving (by <a href="/functor-implication/dual_continuous_consequences">this result</a>).


functor_id: abelianization
property_id: finitary
reason:
Since it is cocontinuous, it is finitary (by <a href="/functor-implication/dual_continuous_consequences">this result</a>).


functor_id: abelianization
property_id: finite-coproduct-preserving
reason:
Since it is coproduct-preserving, it is finite-coproduct-preserving (by <a href="/functor-implication/dual_products_consequences">this result</a>).


functor_id: abelianization
property_id: initial-object-preserving
reason:
Since it is finite-coproduct-preserving, it is initial-object-preserving (by <a href="/functor-implication/dual_finite_products_consequences">this result</a>).


functor_id: abelianization
property_id: right exact
reason:
Since it is coequalizer-preserving and is finite-coproduct-preserving, it is right exact (by <a href="/functor-implication/dual_left_exact_criterion">this result</a>).


functor_id: abelianization
property_id: epimorphism-preserving
reason:
Since it is right exact, it is epimorphism-preserving (by <a href="/functor-implication/dual_left_exact_preserves_mono">this result</a>).


functor_id: abelianization
property_id: equivalence
reason:
Assume for contradiction that it is an equivalence. Then it is faithful (by <a href="/functor-implication/equivalence_criterion">this result</a>) – contradiction.


functor_id: abelianization
property_id: continuous
reason:
Assume for contradiction that it is continuous. Then it is product-preserving (by <a href="/functor-implication/continuous_consequences">this result</a>) – contradiction.


functor_id: abelianization
property_id: cofinitary
reason:
Assume for contradiction that it is cofinitary. Then it is cofinitary and is finite-product-preserving, so it is product-preserving (by <a href="/functor-implication/product_criterion_filtered">this result</a>) – contradiction.


functor_id: abelianization
property_id: right adjoint
reason:
Assume for contradiction that it is a right adjoint. Then it is continuous (by <a href="/functor-implication/right_adjoint_consequences">this result</a>) – contradiction.


functor_id: abelianization
property_id: monadic
reason:
Assume for contradiction that it is monadic. Then it is conservative (by <a href="/functor-implication/monadic_consequences">this result</a>) – contradiction.


functor_id: abelianization
property_id: representable
reason:
Assume for contradiction that it is representable. Then it is continuous (by <a href="/functor-implication/representable_is_continuous">this result</a>) – contradiction.


functor_id: abelianization
property_id: left exact
reason:
Assume for contradiction that it is left exact. Then it is monomorphism-preserving (by <a href="/functor-implication/left_exact_preserves_mono">this result</a>) – contradiction.


functor_id: abelianization
property_id: exact
reason:
Assume for contradiction that it is exact. Then it is left exact (by <a href="/functor-implication/exact_definition">this result</a>) – contradiction.


functor_id: abelianization
property_id: equalizer-preserving
reason:
Assume for contradiction that it is equalizer-preserving. Then it is equalizer-preserving and is finite-product-preserving, so it is left exact (by <a href="/functor-implication/left_exact_criterion">this result</a>) – contradiction.


functor_id: abelianization
property_id: comonadic
reason:
Assume for contradiction that it is comonadic. Then it is conservative (by <a href="/functor-implication/dual_monadic_consequences">this result</a>) – contradiction.


functor_id: power_set_contravariant
property_id: conservative
reason:
Since it is monadic, it is conservative (by <a href="/functor-implication/monadic_consequences">this result</a>).


functor_id: power_set_contravariant
property_id: faithful
reason:
Since it is monadic, it is faithful (by <a href="/functor-implication/monadic_consequences">this result</a>).


functor_id: power_set_contravariant
property_id: right adjoint
reason:
Since it is monadic, it is a right adjoint (by <a href="/functor-implication/monadic_consequences">this result</a>).


functor_id: power_set_contravariant
property_id: continuous
reason:
Since it is a right adjoint, it is continuous (by <a href="/functor-implication/right_adjoint_consequences">this result</a>).


functor_id: power_set_contravariant
property_id: cofinitary
reason:
Since it is continuous, it is cofinitary (by <a href="/functor-implication/continuous_consequences">this result</a>).


functor_id: power_set_contravariant
property_id: equalizer-preserving
reason:
Since it is continuous, it is equalizer-preserving (by <a href="/functor-implication/continuous_consequences">this result</a>).


functor_id: power_set_contravariant
property_id: product-preserving
reason:
Since it is continuous, it is product-preserving (by <a href="/functor-implication/continuous_consequences">this result</a>).


functor_id: power_set_contravariant
property_id: finite-product-preserving
reason:
Since it is product-preserving, it is finite-product-preserving (by <a href="/functor-implication/products_consequences">this result</a>).


functor_id: power_set_contravariant
property_id: left exact
reason:
Since it is equalizer-preserving and is finite-product-preserving, it is left exact (by <a href="/functor-implication/left_exact_criterion">this result</a>).


functor_id: power_set_contravariant
property_id: monomorphism-preserving
reason:
Since it is left exact, it is monomorphism-preserving (by <a href="/functor-implication/left_exact_preserves_mono">this result</a>).


functor_id: power_set_contravariant
property_id: equivalence
reason:
Assume for contradiction that it is an equivalence. Then it is essentially surjective (by <a href="/functor-implication/equivalence_criterion">this result</a>) – contradiction.


functor_id: power_set_contravariant
property_id: cocontinuous
reason:
Assume for contradiction that it is cocontinuous. Then it is coequalizer-preserving (by <a href="/functor-implication/dual_continuous_consequences">this result</a>) – contradiction.


functor_id: power_set_contravariant
property_id: finite-coproduct-preserving
reason:
Assume for contradiction that it is finite-coproduct-preserving. Then it is initial-object-preserving (by <a href="/functor-implication/dual_finite_products_consequences">this result</a>) – contradiction.


functor_id: power_set_contravariant
property_id: coproduct-preserving
reason:
Assume for contradiction that it is coproduct-preserving. Then it is finite-coproduct-preserving (by <a href="/functor-implication/dual_products_consequences">this result</a>) – contradiction.


functor_id: power_set_contravariant
property_id: right exact
reason:
Assume for contradiction that it is right exact. Then it is finite-coproduct-preserving (by <a href="/functor-implication/dual_left_exact_consequences">this result</a>) – contradiction.


functor_id: power_set_contravariant
property_id: exact
reason:
Assume for contradiction that it is exact. Then it is right exact (by <a href="/functor-implication/exact_definition">this result</a>) – contradiction.


functor_id: power_set_contravariant
property_id: left adjoint
reason:
Assume for contradiction that it is a left adjoint. Then it is cocontinuous (by <a href="/functor-implication/dual_right_adjoint_consequences">this result</a>) – contradiction.


functor_id: power_set_contravariant
property_id: comonadic
reason:
Assume for contradiction that it is comonadic. Then it is a left adjoint (by <a href="/functor-implication/dual_monadic_consequences">this result</a>) – contradiction.


functor_id: power_set_covariant
property_id: continuous
reason:
Assume for contradiction that it is continuous. Then it is equalizer-preserving (by <a href="/functor-implication/continuous_consequences">this result</a>) – contradiction.


functor_id: power_set_covariant
property_id: finite-product-preserving
reason:
Assume for contradiction that it is finite-product-preserving. Then it is terminal-object-preserving (by <a href="/functor-implication/finite_products_consequences">this result</a>) – contradiction.


functor_id: power_set_covariant
property_id: product-preserving
reason:
Assume for contradiction that it is product-preserving. Then it is finite-product-preserving (by <a href="/functor-implication/products_consequences">this result</a>) – contradiction.


functor_id: power_set_covariant
property_id: left exact
reason:
Assume for contradiction that it is left exact. Then it is finite-product-preserving (by <a href="/functor-implication/left_exact_consequences">this result</a>) – contradiction.


functor_id: power_set_covariant
property_id: monadic
reason:
Assume for contradiction that it is monadic. Then it is a right adjoint (by <a href="/functor-implication/monadic_consequences">this result</a>) – contradiction.


functor_id: forget_vector
property_id: full
reason:
Assume for contradiction that it is full. Then it is full and is monadic, so it is an equivalence (by <a href="/functor-implication/monadic_full_trivial">this result</a>) – contradiction.


functor_id: forget_vector
property_id: right exact
reason:
Assume for contradiction that it is right exact. Then it is finitary and is right exact, so it is cocontinuous (by <a href="/functor-implication/dual_continuous_criterion_filtered">this result</a>) – contradiction.


functor_id: id_Set
property_id: essentially surjective
reason:
Since it is an equivalence, it is essentially surjective (by <a href="/functor-implication/equivalence_criterion">this result</a>).


functor_id: id_Set
property_id: faithful
reason:
Since it is an equivalence, it is faithful (by <a href="/functor-implication/equivalence_criterion">this result</a>).


functor_id: id_Set
property_id: full
reason:
Since it is an equivalence, it is full (by <a href="/functor-implication/equivalence_criterion">this result</a>).


functor_id: id_Set
property_id: continuous
reason:
Since it is an equivalence, it is continuous (by <a href="/functor-implication/equivalence_consequences">this result</a>).


functor_id: id_Set
property_id: monadic
reason:
Since it is an equivalence, it is monadic (by <a href="/functor-implication/equivalence_consequences">this result</a>).


functor_id: id_Set
property_id: right adjoint
reason:
Since it is an equivalence, it is a right adjoint (by <a href="/functor-implication/equivalence_consequences">this result</a>).


functor_id: id_Set
property_id: cocontinuous
reason:
Since it is an equivalence, it is cocontinuous (by <a href="/functor-implication/dual_equivalence_consequences">this result</a>).


functor_id: id_Set
property_id: exact
reason:
Since it is left exact and is right exact, it is exact (by <a href="/functor-implication/exact_definition">this result</a>).


functor_id: id_Set
property_id: comonadic
reason:
Since it is an equivalence, it is comonadic (by <a href="/functor-implication/dual_equivalence_consequences">this result</a>).


functor_id: id_Set
property_id: left adjoint
reason:
Since it is an equivalence, it is a left adjoint (by <a href="/functor-implication/dual_equivalence_consequences">this result</a>).


