category of finite sets
- notation:
- objects: finite sets
- morphisms: maps
- Related categories:
- nLab Link
Satisfied Properties
Properties from the database
- has a cogenerator
- is an elementary topos
- is essentially small
- has a generator
- is locally small
- is strongly connected
Deduced properties
- is locally essentially small
- is well-copowered
- is well-powered
- has a generating set
- is inhabited
- is cartesian closed
- has finite products
- has binary products
- has a terminal object
- is connected
- has finite powers
- has binary powers
- is finitely complete
- has equalizers
- has coreflexive equalizers
- has pullbacks
- is Cauchy complete
- has a subobject classifier
- is mono-regular
- is balanced
- has a regular subobject classifier
- has disjoint finite coproducts
- has finite coproducts
- is distributive
- has a strict initial object
- has an initial object
- is epi-regular
- is finitely cocomplete
- is locally cartesian closed
- is coregular
- is extensive
- is lextensive
- is co-Malcev
- has binary coproducts
- has coequalizers
- is regular
- has reflexive coequalizers
- has pushouts
- has finite copowers
- has binary copowers
- has a cogenerating set
Unsatisfied Properties
Properties from the database
- does not have countable copowers
- does not have countable powers
- is not Malcev
- is not skeletal
- is not small
- does not have a strict terminal object
Deduced properties*
- does not have countable products
- does not have products
- does not have cofiltered limits
- is not complete
- does not have wide pullbacks
- does not have connected limits
- is not essentially finite
- does not have sequential limits
- does not have directed limits
- does not have powers
- does not have copowers
- is not finite
- is not a groupoid
- is not direct
- is not discrete
- is not essentially discrete
- is not trivial
- is not thin
- is not pointed
- does not have zero morphisms
- does not have biproducts
- is not left cancellative
- is not one-way
- is not preadditive
- is not additive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not locally presentable
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not locally strongly finitely presentable
- is not finitary algebraic
- is not a Grothendieck topos
- does not have coproducts
- does not have disjoint coproducts
- is not infinitary distributive
- is not infinitary extensive
- is not cocomplete
- is not unital
- does not have cosifted limits
- does not have filtered colimits
- does not have sifted colimits
- does not have connected colimits
- does not have directed colimits
- does not have exact filtered colimits
- does not have wide pushouts
- does not have countable coproducts
- does not have sequential colimits
- does not have disjoint products
- does not have disjoint finite products
- is not infinitary codistributive
- is not right cancellative
- is not infinitary coextensive
- is not codistributive
- is not coextensive
- is not inverse
- is not counital
- is not self-dual
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- terminal object: singleton set
- initial object: empty set
- products: [finite case] direct products
- coproducts: [finite case] disjoint union
Special morphisms
- isomorphisms: bijective maps
- monomorphisms: injective maps
- epimorphisms: surjective maps
- regular monomorphisms: same as monomorphisms
- regular epimorphisms: same as epimorphisms
Comments
- For the non-existence of sequential (co-)limits it is not sufficient to take a diagram of finite sets whose (co-)limit in is not contained in .