category of non-empty sets
- notation:
- objects: non-empty sets
- morphisms: maps
- nLab Link
- Related categories:
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.
Properties
Properties from the database
- is cartesian closed
- has coequalizers
- has a cogenerator
- is epi-regular
- has a generator
- is locally small
- is mono-regular
- has products
- is well-copowered
- is well-powered
- has wide pushouts
Deduced properties
- is locally essentially small
- has countable products
- has finite products
- has binary products
- has a terminal object
- is connected
- is inhabited
- is balanced
- has connected colimits
- has filtered colimits
- has pushouts
- is Cauchy complete
- has sequential colimits
Non-Properties
Non-Properties from the database
- does not have binary coproducts
- does not have sequential limits
- is not skeletal
- does not have a strict terminal object
Deduced Non-Properties*
- is not discrete
- does not have equalizers
- is not thin
- is not essentially small
- is not small
- is not essentially finite
- is not finite
- is not essentially discrete
- is not trivial
- is not left cancellative
- is not complete
- is not finitely complete
- does not have pullbacks
- is not pointed
- does not have connected limits
- does not have wide pullbacks
- does not have filtered limits
- does not have exact filtered colimits
- is not locally presentable
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not finitary algebraic
- is not an elementary topos
- is not a Grothendieck topos
- does not have a subobject classifier
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not a groupoid
- is not Malcev
- does not have finite coproducts
- does not have disjoint finite coproducts
- does not have disjoint coproducts
- is not distributive
- is not infinitary distributive
- does not have coproducts
- is not additive
- is not preadditive
- is not cocomplete
- is not finitely cocomplete
- does not have an initial object
- does not have a strict initial object
- does not have zero morphisms
- is not right cancellative
- does not have countable coproducts
- is not self-dual
*This also uses the deduced properties.
Unknown properties
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Special morphisms
- Isomorphisms: bijective maps
- Monomorphisms: injective maps
- Epimorphisms: surjective maps