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
Assigned properties
Deduced properties
- is locally strongly finitely presentable
- is well-copowered
- is connected
- has a generator
- is locally essentially small
- has coproducts
- is an elementary topos
- has a generating set
- has a cogenerator
- has exact filtered colimits
- is infinitary extensive
- is locally presentable
- is accessible
- is cocomplete
- is locally finitely presentable
- is complete
- is a generalized variety
- is regular
- is multi-algebraic
- is inhabited
- has filtered colimits
- is finitely complete
- has filtered-colimit-stable monomorphisms
- has cartesian filtered colimits
- is extensive
- is cartesian closed
- has a subobject classifier
- has disjoint finite coproducts
- has effective congruences
- is epi-regular
- is finitely cocomplete
- is locally cartesian closed
- has a cogenerating set
- has copowers
- has ℵ₂-small coproducts
- is finitely accessible
- is locally ℵ₁-presentable
- is well-powered
- is Cauchy complete
- has sifted colimits
- is ℵ₁-accessible
- is locally finitely multi-presentable
- is multi-cocomplete
- has finite products
- has powers
- has pullbacks
- has connected limits
- has equalizers
- has products
- is multi-complete
- has quotients of congruences
- is co-Malcev
- has effective cocongruences
- is mono-regular
- is Barr-exact
- has disjoint coproducts
- has finite coproducts
- is infinitary distributive
- has a strict initial object
- is filtered
- has directed colimits
- has ℵ₁-filtered colimits
- has a regular subobject classifier
- has connected colimits
- has coequalizers
- is cofiltered
- is balanced
- has countable coproducts
- has ℵ₂-small copowers
- is locally multi-presentable
- has a multi-terminal object
- is countably distributive
- is distributive
- has coreflexive equalizers
- is sifted
- is ℵ₁-filtered
- has reflexive coequalizers
- has an initial object
- has ℵ₂-small products
- has binary products
- has a terminal object
- has finite powers
- has ℵ₂-small powers
- has wide pullbacks
- is a pretopos
- is a quasitopos
- has a multi-initial object
- is cosifted
- has sequential colimits
- has cosifted limits
- has binary coproducts
- has countable copowers
- has finite copowers
- has wide pushouts
- has a natural numbers object
- is locally poly-presentable
- has countable powers
- has countable products
- has binary powers
- has cofiltered limits
- is coregular
- has coquotients of cocongruences
- is ℵ₁-cofiltered
- has binary copowers
- has pushouts
- has cocartesian cofiltered limits
- has sequential limits
- is Barr-coexact
- has directed limits
- has ℵ₁-cofiltered limits
Unsatisfied Properties
Assigned properties
Deduced properties*
- is not pointed
- is not discrete
- is not essentially discrete
- is not thin
- is not additive
- is not regular-subobject-trivial
- is not regular-quotient-trivial
- is not gaunt
- is not direct
- does not have cofiltered-limit-stable epimorphisms
- is not inverse
- is not unital
- is not preadditive
- is not abelian
- is not left cancellative
- does not have a strict terminal object
- is not strongly connected
- is not a groupoid
- does not have zero morphisms
- is not subobject-trivial
- is not right cancellative
- is not Malcev
- is not core-thin
- is not locally finite
- is not essentially small
- is not essentially countable
- is not essentially finite
- is not counital
- is not cocartesian coclosed
- does not have disjoint finite products
- does not have exact cofiltered limits
- is not quotient-trivial
- does not have a regular quotient object classifier
- is not self-dual
- does not have biproducts
- is not locally copresentable
- is not Grothendieck abelian
- is not split abelian
- does not have kernels
- does not satisfy CIP
- is not normal
- is not small
- is not finite
- is not countable
- is not one-way
- is not locally cocartesian coclosed
- is not coextensive
- does not have disjoint products
- is not codistributive
- does not have cokernels
- does not satisfy CSP
- is not conormal
- does not have a quotient object classifier
- is not coaccessible
- is not countably codistributive
- is not infinitary coextensive
- is not infinitary codistributive
*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
Functors
There are 19 functors whose source or target is the category of sets.
- binary coproduct functor on sets
- binary diagonal functor on the category of sets
- binary product functor on sets
- contravariant power set functor
- countable copower functor on sets
- covariant power set functor
- discrete topology functor
- doubling functor on sets
- forgetful functor for groups
- forgetful functor for rings
- forgetful functor for topological spaces
- forgetful functor for vector spaces
- free group functor
- identity functor on the category of sets
- indiscrete topology functor
- path components functor
- sequences functor on sets
- squaring functor on sets
- trivial functor from the category of sets