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 powers
- has binary products
- has a terminal object
- has countable powers
- has pullbacks
- is Cauchy complete
- has wide pullbacks
- has cofiltered limits
- has sequential limits
- has finite powers
- has binary powers
- is multi-complete
- has a multi-terminal object
- is connected
- has filtered colimits
- has directed colimits
- has cartesian filtered colimits
- has coproducts
- is extensive
- has finite coproducts
- has cocartesian cofiltered limits
- has disjoint finite coproducts
- has disjoint coproducts
- has a strict initial object
- has an initial object
- is distributive
- is infinitary distributive
- is countably distributive
- has countable coproducts
- is balanced
- has a natural numbers object
- is inhabited
- is filtered
- is sifted
- has connected colimits
- has sifted colimits
- has reflexive coequalizers
- is finitely cocomplete
- has cosifted limits
- has copowers
- has binary coproducts
- has coequalizers
- has countable copowers
- has sequential colimits
- has pushouts
- has directed limits
- has wide pushouts
- has finite copowers
- has binary copowers
- is multi-cocomplete
- has a multi-initial object
- is cosifted
- is cofiltered
Unsatisfied Properties
Properties from the database
- is not cartesian closed
- is not locally essentially small
- is not Malcev
- is not semi-strongly connected
- is not skeletal
- is not well-powered
Deduced properties*
- is not essentially small
- is not small
- is not essentially countable
- is not essentially finite
- is not finite
- is not countable
- is not locally small
- is not a groupoid
- is not direct
- is not thin
- does not have a strict terminal object
- is not right cancellative
- is not left cancellative
- is not one-way
- is not preadditive
- is not additive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not discrete
- is not essentially discrete
- is not trivial
- is not pointed
- does not have zero morphisms
- does not have kernels
- does not have biproducts
- is not normal
- is not accessible
- is not locally presentable
- is not locally ℵ₁-presentable
- is not locally finitely presentable
- is not locally strongly finitely presentable
- is not finitary algebraic
- is not finitely accessible
- is not ℵ₁-accessible
- is not a generalized variety
- is not locally multi-presentable
- is not locally finitely multi-presentable
- is not locally poly-presentable
- is not multi-algebraic
- is not an elementary topos
- is not locally cartesian closed
- is not a Grothendieck topos
- is not strongly connected
- is not unital
- does not have cokernels
- does not have disjoint finite products
- does not have disjoint products
- is not codistributive
- is not countably codistributive
- is not infinitary codistributive
- is not cocartesian coclosed
- is not coextensive
- is not infinitary coextensive
- is not conormal
- is not inverse
- is not coaccessible
- is not locally copresentable
- does not have a regular quotient object classifier
- does not have a quotient object classifier
- is not locally cocartesian coclosed
- is not counital
- is not self-dual
*This also uses the deduced satisfied properties.
Unknown properties
There are 7 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
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