dual of the category of sets
- notation:
- objects: sets
- morphisms: A morphism is a map of sets .
- nLab Link
- Dual category:
By definition, this category is the dual (or opposite) of the category .
Satisfied Properties
Assigned properties
—
Deduced properties
- is locally small
- is semi-strongly connected
- is well-powered
- is connected
- has a cogenerator
- is locally essentially small
- has products
- has a cogenerating set
- has a generator
- has exact cofiltered limits
- is infinitary coextensive
- is locally copresentable
- is coaccessible
- is complete
- is cocomplete
- is coregular
- is inhabited
- has cofiltered limits
- is finitely cocomplete
- has cofiltered-limit-stable epimorphisms
- has cocartesian cofiltered limits
- is coextensive
- is cocartesian coclosed
- has a quotient object classifier
- has disjoint finite products
- has effective cocongruences
- is mono-regular
- is finitely complete
- is regular
- is locally cocartesian coclosed
- has a generating set
- has powers
- has countable products
- is well-copowered
- is Cauchy complete
- has cosifted limits
- is multi-complete
- has finite coproducts
- has copowers
- has pushouts
- has connected colimits
- has coequalizers
- has coproducts
- is multi-cocomplete
- has coquotients of cocongruences
- is Malcev
- has effective congruences
- is epi-regular
- is Barr-coexact
- has disjoint products
- has finite products
- is countably codistributive
- is infinitary codistributive
- has a strict terminal object
- is cofiltered
- has directed limits
- has a regular quotient object classifier
- has connected limits
- has equalizers
- is filtered
- is balanced
- has countable powers
- has countable copowers
- has a multi-initial object
- is codistributive
- has reflexive coequalizers
- is cosifted
- has coreflexive equalizers
- has a terminal object
- has countable coproducts
- has binary coproducts
- has an initial object
- has finite copowers
- has wide pushouts
- has a multi-terminal object
- has quotients of congruences
- is Barr-exact
- is sifted
- has sequential limits
- has sifted colimits
- has binary products
- has finite powers
- has wide pullbacks
- has sequential colimits
- has binary copowers
- has filtered colimits
- has binary powers
- has pullbacks
- has cartesian filtered colimits
- has directed colimits
Unsatisfied Properties
Assigned properties
—
Deduced properties*
- is not skeletal
- is not trivial
- is not pointed
- is not discrete
- is not essentially discrete
- is not thin
- is not additive
- is not gaunt
- is not inverse
- does not have filtered-colimit-stable monomorphisms
- is not direct
- is not counital
- is not preadditive
- is not abelian
- is not left cancellative
- is not right cancellative
- does not have a strict initial object
- is not strongly connected
- is not a groupoid
- does not have zero morphisms
- is not quotient-trivial
- is not core-thin
- is not locally finite
- is not essentially small
- is not essentially countable
- is not essentially finite
- is not unital
- is not cartesian closed
- does not have disjoint finite coproducts
- does not have exact filtered colimits
- is not subobject-trivial
- does not have a regular subobject classifier
- is not self-dual
- does not have biproducts
- is not locally presentable
- is not split abelian
- does not have cokernels
- does not satisfy CSP
- is not conormal
- is not small
- is not finite
- is not countable
- is not co-Malcev
- is not one-way
- is not locally cartesian closed
- does not have disjoint coproducts
- is not distributive
- does not have kernels
- does not satisfy CIP
- is not extensive
- is not normal
- does not have a subobject classifier
- is not accessible
- is not countably distributive
- is not infinitary extensive
- is not infinitary distributive
- does not have a natural numbers object
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not Grothendieck abelian
- is not ℵ₁-accessible
- is not locally multi-presentable
- is not locally poly-presentable
- is not locally finitely multi-presentable
- is not an elementary topos
- is not a Grothendieck topos
- is not finitely accessible
- is not locally strongly finitely presentable
- is not a generalized variety
- is not multi-algebraic
- is not finitary algebraic
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- terminal object: empty set
- initial object: singleton set
- products: disjoint union
- coproducts: direct products with pointwise operations
Special morphisms
- isomorphisms: bijective maps
- monomorphisms: surjective maps
- epimorphisms: injective maps
- regular monomorphisms: surjective homomorphisms
- regular epimorphisms: same as monomorphisms