category of sets
- notation:
- objects: sets
- morphisms: maps
- Related categories: , ,
- nLab Link
- Dual category:
The category of sets plays a fundamental role in category theory. Due to the Yoneda embedding, many results about general categories can be reduced to the category of sets. It is also usually the first example of a category that one encounters.
Satisfied Properties
Properties from the database
- is finitary algebraic
- is a Grothendieck topos
- is locally small
- is strongly connected
Deduced properties
- is locally essentially small
- has a generator
- has a generating set
- is inhabited
- is locally strongly finitely presentable
- is well-copowered
- is regular
- is finitely complete
- has equalizers
- has coreflexive equalizers
- has finite products
- has binary products
- has a terminal object
- is connected
- has pullbacks
- is Cauchy complete
- has finite powers
- has binary powers
- is locally finitely presentable
- is locally presentable
- is locally ℵ₁-presentable
- is cocomplete
- is complete
- has connected limits
- has products
- has countable products
- has wide pullbacks
- has cofiltered limits
- has sequential limits
- has powers
- has countable powers
- is well-powered
- has exact filtered colimits
- has filtered colimits
- has directed colimits
- has coproducts
- is an elementary topos
- is cartesian closed
- is infinitary distributive
- is distributive
- has finite coproducts
- has a strict initial object
- has an initial object
- has a subobject classifier
- is mono-regular
- is balanced
- has a regular subobject classifier
- has disjoint finite coproducts
- has disjoint coproducts
- is epi-regular
- is finitely cocomplete
- has a cogenerator
- is locally cartesian closed
- is coregular
- is extensive
- is lextensive
- is infinitary extensive
- is co-Malcev
- has connected colimits
- has sifted colimits
- has reflexive coequalizers
- has cosifted limits
- has countable coproducts
- has binary coproducts
- has coequalizers
- has sequential colimits
- has pushouts
- has directed limits
- has wide pushouts
- has copowers
- has countable copowers
- has finite copowers
- has binary copowers
- has a cogenerating set
Unsatisfied Properties
Properties from the database
- is not Malcev
- is not skeletal
- does not have a strict terminal object
Deduced properties*
- is not direct
- is not discrete
- is not thin
- is not left cancellative
- is not a groupoid
- is not one-way
- is not essentially discrete
- is not trivial
- is not pointed
- does not have zero morphisms
- does not have biproducts
- is not preadditive
- is not additive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not essentially small
- is not small
- is not finite
- is not essentially finite
- is not self-dual
- is not unital
- does not have disjoint finite products
- does not have disjoint products
- is not right cancellative
- is not codistributive
- is not infinitary codistributive
- is not coextensive
- is not infinitary coextensive
- is not inverse
- is not counital
*This also uses the deduced satisfied properties.
Unknown properties
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Special objects
- terminal object: singleton set
- initial object: empty set
- products: direct products with pointwise operations
- coproducts: disjoint union
Special morphisms
- isomorphisms: bijective maps
- monomorphisms: injective maps
- epimorphisms: surjective maps
- regular monomorphisms: same as monomorphisms
- regular epimorphisms: surjective homomorphisms