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.
Satisfied Properties
Properties from the database
- is co-Malcev
- is cocomplete
- is complete
- is coregular
- is epi-regular
- has exact filtered colimits
- is infinitary extensive
- is mono-regular
- is regular
Deduced properties
- has connected limits
- is finitely complete
- has equalizers
- has coreflexive equalizers
- has products
- has countable products
- has finite products
- has binary products
- has a terminal object
- is connected
- has pullbacks
- is Cauchy complete
- has wide pullbacks
- has cofiltered limits
- has sequential limits
- has powers
- has countable powers
- has finite powers
- has binary powers
- has filtered colimits
- has directed colimits
- has coproducts
- is extensive
- has finite coproducts
- has a strict initial object
- has an initial object
- has disjoint finite coproducts
- has disjoint coproducts
- is distributive
- is infinitary distributive
- is lextensive
- is balanced
- is inhabited
- has connected colimits
- has sifted colimits
- has reflexive coequalizers
- is finitely cocomplete
- 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
Unsatisfied Properties
Properties from the database
- is not cartesian closed
- is not locally essentially small
- is not Malcev
- is not skeletal
- does not have a strict terminal object
- is not strongly connected
Deduced properties*
- is not essentially small
- is not small
- is not finite
- is not essentially finite
- is not locally small
- is not a groupoid
- is not direct
- is not discrete
- does not have zero morphisms
- does not have biproducts
- is not pointed
- is not preadditive
- is not additive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not essentially discrete
- is not trivial
- is not thin
- is not left cancellative
- is not one-way
- is not locally presentable
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not locally strongly finitely presentable
- is not finitary algebraic
- is not an elementary topos
- is not locally cartesian closed
- is not a Grothendieck topos
- 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
- is not self-dual
*This also uses the deduced satisfied properties.
Unknown properties
There are 8 properties for which the database doesn't have an answer if they are satisfied or not. Please help to contribute the data!
- has a cogenerating set
- has a cogenerator
- has a generating set
- has a generator
- has a regular subobject classifier
- has a subobject classifier
- is well-copowered
- is well-powered
Special objects
- terminal object: constant functor with value
- initial object: constant functor with value
- products: pointwise defined direct product
- coproducts: pointwise disjoint union
Special morphisms
- isomorphisms: natural isomorphisms
- monomorphisms: pointwise injective natural transformations
- epimorphisms: objectwise surjective natural transformations
- regular monomorphisms: same as monomorphisms
- regular epimorphisms: same as epimorphisms