real interval [0,1]
Every poset can be regarded as a small thin category. This is a specific example. This category is locally -presentable (in fact, every object is -presentable), but not locally finitely presentable (in fact, only is finitely presentable).
Satisfied properties
Assigned properties
- is small
- is self-dual
- is skeletal
- is semi-strongly connected
- is locally ℵ₁-presentable
Deduced properties
- is cocomplete
- is ℵ₁-accessible
- is connected
- is essentially small
- is locally small
- is locally ℵ₁-copresentable
- is accessible
- has ℵ₁-filtered colimits
- is inhabited
- has an extremal generating collection
- is locally essentially small
- is well-powered
- is complete
- is ℵ₁-coaccessible
- has connected colimits
- has coproducts
- is finitely cocomplete
- is multi-cocomplete
- has an extremal cogenerating collection
- is well-copowered
- is locally presentable
- is Cauchy complete
- is locally multi-presentable
- has connected limits
- is finitely complete
- has products
- is multi-complete
- is filtered
- has sifted colimits
- has a generating collection
- is coaccessible
- has ℵ₁-cofiltered limits
- has coequalizers
- has finite coproducts
- has a multi-initial object
- has a cogenerating collection
- has copowers
- has ℵ₂-small coproducts
- has wide pushouts
- has equalizers
- has finite products
- has a multi-terminal object
- is sifted
- has filtered colimits
- has reflexive coequalizers
- has powers
- has ℵ₂-small products
- has wide pullbacks
- is concretizable
- is total
- is locally copresentable
- is locally multi-copresentable
- is locally poly-copresentable
- has an initial object
- is cofiltered
- has cosifted limits
- has countable coproducts
- has ℵ₂-small copowers
- has binary coproducts
- has finite copowers
- has pushouts
- is thin
- is cototal
- is Malcev
- is locally poly-presentable
- is locally cartesian closed
- has a terminal object
- has quotients of congruences
- has coreflexive equalizers
- is ℵ₁-filtered
- has directed colimits
- has countable products
- has ℵ₂-small powers
- has binary products
- has finite powers
- has cofiltered limits
- has pullbacks
- is left cancellative
- is locally finite
- is one-way
- has a generator
- has binary powers
- is co-Malcev
- is locally cocartesian coclosed
- is cosifted
- is ℵ₁-cofiltered
- has sequential colimits
- has countable copowers
- has binary copowers
- has cokernel pairs
- is right cancellative
- has a cogenerator
- has a natural numbers object
- has a parametrized natural numbers object
- has kernel pairs
- has a strict initial object
- is cartesian closed
- has pullback-stable regular epimorphisms
- has filtered-colimit-stable monomorphisms
- has sequential limits
- has countable powers
- has a regular subobject classifier
- is regular-subobject-trivial
- is retract-trivial
- is core-thin
- has a coparametrized conatural numbers object
- has a strict terminal object
- is cocartesian coclosed
- has pushout-stable regular monomorphisms
- has coquotients of cocongruences
- has cofiltered-limit-stable epimorphisms
- has directed limits
- has a regular quotient object classifier
- is regular-quotient-trivial
- has cartesian filtered colimits
- has coequalizers of kernel pairs
- has effective congruences
- is distributive
- is countably distributive
- is infinitary distributive
- is gaunt
- is a quasitopos
- has a conatural numbers object
- has cocartesian cofiltered limits
- has equalizers of cokernel pairs
- has effective cocongruences
- is codistributive
- is countably codistributive
- is infinitary codistributive
- is a quasicotopos
- has exact filtered colimits
- is regular
- is coregular
- has exact cofiltered limits
- is Barr-exact
- is Barr-coexact
Unsatisfied properties
Assigned properties
- is not essentially countable
- does not have an extremal generator
- is not locally finitely presentable
Deduced properties*
- is not finitely accessible
- is not finitary algebraic
- is not core-connected
- is not balanced
- is not strongly connected
- is not one-sorted finitary algebraic
- is not essentially finite
- is not countable
- does not have an extremal cogenerator
- is not locally finitely copresentable
- is not a generalized variety
- is not locally finitely multi-presentable
- is not a groupoid
- does not have zero morphisms
- is not trivial
- is not mono-regular
- is not finite
- is not finitely coaccessible
- is not dually finitary algebraic
- is not inverse
- is not dually one-sorted finitary algebraic
- is not epi-regular
- is not pointed
- is not preadditive
- does not have biproducts
- is not direct
- is not multi-algebraic
- is not extensive
- is not essentially discrete
- does not have kernels
- does not satisfy CIP
- is not infinitary extensive
- is not normal
- does not have a subobject classifier
- is not subobject-trivial
- does not have disjoint finite coproducts
- is not a dual generalized variety
- is not locally finitely multi-copresentable
- is not coextensive
- is not discrete
- does not have cokernels
- does not satisfy CSP
- is not infinitary coextensive
- is not conormal
- does not have a quotient object classifier
- is not quotient-trivial
- does not have disjoint finite products
- is not unital
- is not additive
- is not abelian
- does not have disjoint coproducts
- is not countably extensive
- is not an elementary topos
- is not a Grothendieck topos
- is not a pretopos
- is not counital
- is not dually multi-algebraic
- does not have disjoint products
- is not countably coextensive
- is not an elementary cotopos
- is not a Grothendieck cotopos
- is not a precotopos
- is not Grothendieck abelian
- is not split abelian
- is not co-Grothendieck abelian
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- terminal object:
- initial object:
- products: infimum
- coproducts: supremum
Special morphisms
- isomorphisms: only the identity morphisms
- monomorphisms: every morphism
- epimorphisms: every morphism
- regular monomorphisms: same as isomorphisms
- regular epimorphisms: same as isomorphisms