category of Z-functors
- notation:
- objects: Z-functors, i.e. functors from commutative rings to sets
- morphisms: natural transformations
- Related categories: ,
This category is used in functorial algebraic geometry. It also provides a typical example of a functor category that is not locally small, but nevertheless relevant. Most of its properties are directly derived from the category of sets, so other functor categories for large categories will be similar.
Properties
Properties from the database
- is cocomplete
- is complete
- has disjoint coproducts
- is epi-regular
- has exact filtered colimits
- is infinitary distributive
- is mono-regular
Deduced properties
- has connected limits
- has filtered limits
- is finitely complete
- has wide pullbacks
- has equalizers
- has products
- has finite products
- has countable products
- has binary products
- has a terminal object
- has pullbacks
- is connected
- has coproducts
- has disjoint finite coproducts
- has finite coproducts
- is Cauchy complete
- has sequential limits
- has filtered colimits
- is distributive
- has a strict initial object
- has an initial object
- is inhabited
- is balanced
- has connected colimits
- is finitely cocomplete
- has wide pushouts
- has coequalizers
- has countable coproducts
- has binary coproducts
- has pushouts
- has sequential colimits
Non-Properties
Non-Properties from the database
- is not Malcev
- is not locally essentially small
- is not skeletal
- does not have a strict terminal object
Deduced Non-Properties*
- is not essentially small
- is not small
- is not finite
- is not essentially finite
- is not locally small
- is not discrete
- is not essentially discrete
- is not trivial
- is not thin
- is not left cancellative
- is not pointed
- does not have zero morphisms
- is not locally presentable
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not finitary algebraic
- is not a Grothendieck topos
- is not preadditive
- is not additive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not a groupoid
- is not right cancellative
- is not self-dual
*This also uses the deduced properties.
Unknown properties
For these properties the database currently doesn't have an answer if they are satisfied or not. Please help to complete the data!
- is cartesian closed
- has a cogenerator
- is an elementary topos
- has a generator
- has a subobject classifier
- is well-copowered
- is well-powered
Special morphisms
- Isomorphisms: natural isomorphisms
- Monomorphisms: pointwise injective natural transformations
- Epimorphisms: objectwise surjective natural transformations
Comments
- Lots of properties are unknown here. Please help to fill in the gaps!