category of non-empty sets
- notation:
- objects: non-empty sets
- morphisms: maps
- Related categories:
- nLab Link
This entry demonstrates that removing an object (the empty set) can drastically change the properties of a category. In particular, this category is neither complete nor cocomplete.
Satisfied Properties
Properties from the database
- has binary coproducts
- is cartesian closed
- has coequalizers
- has a cogenerator
- has disjoint finite products
- is epi-regular
- has a generator
- is locally small
- is mono-regular
- has products
- is strongly connected
- is well-copowered
- is well-powered
- has wide pushouts
Deduced properties
- has countable products
- has finite products
- has binary products
- has a terminal object
- is connected
- has powers
- has countable powers
- has finite powers
- has binary powers
- is locally essentially small
- has a generating set
- is inhabited
- is balanced
- has reflexive coequalizers
- has pushouts
- is Cauchy complete
- has connected colimits
- has sifted colimits
- has filtered colimits
- has directed colimits
- has sequential colimits
- has binary copowers
- has disjoint products
- has a cogenerating set
Unsatisfied Properties
Properties from the database
- does not have an initial object
- does not have sequential limits
- is not skeletal
- does not have a strict terminal object
Deduced properties*
- does not have directed limits
- does not have equalizers
- does not have coreflexive equalizers
- is not complete
- is not finitely complete
- does not have pullbacks
- does not have connected limits
- does not have wide pullbacks
- is not pointed
- does not have exact filtered colimits
- does not have a strict initial object
- is not distributive
- is not infinitary distributive
- does not have finite coproducts
- does not have biproducts
- does not have disjoint finite coproducts
- does not have disjoint coproducts
- is not extensive
- is not infinitary extensive
- does not have coproducts
- is not regular
- is not lextensive
- is not left cancellative
- is not a groupoid
- is not direct
- is not discrete
- is not additive
- is not preadditive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not locally strongly finitely presentable
- is not finitary algebraic
- is not essentially discrete
- is not trivial
- is not thin
- is not one-way
- is not essentially small
- is not small
- is not finite
- is not essentially finite
- is not locally presentable
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not an elementary topos
- does not have a subobject classifier
- does not have a regular subobject classifier
- is not locally cartesian closed
- is not a Grothendieck topos
- is not Malcev
- is not unital
- does not have cosifted limits
- does not have countable coproducts
- is not cocomplete
- is not finitely cocomplete
- does not have cofiltered limits
- does not have finite copowers
- does not have countable copowers
- does not have copowers
- does not have zero morphisms
- is not infinitary codistributive
- is not codistributive
- is not right cancellative
- is not coextensive
- is not infinitary coextensive
- is not coregular
- is not inverse
- is not co-Malcev
- is not counital
- is not self-dual
*This also uses the deduced satisfied properties.
Unknown properties
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Special objects
- terminal object: singleton set
- products: direct products
Special morphisms
- isomorphisms: bijective maps
- monomorphisms: injective maps
- epimorphisms: surjective maps
- regular monomorphisms: same as monomorphisms
- regular epimorphisms: same as epimorphisms