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
Assigned properties
- is complete
- is cocomplete
- is infinitary extensive
- has exact filtered colimits
- is regular
- is coregular
- is co-Malcev
- has effective congruences
- has effective cocongruences
Deduced properties
- has connected limits
- is finitely complete
- has equalizers
- has products
- is multi-complete
- has quotients of congruences
- is Barr-exact
- has filtered colimits
- has filtered-colimit-stable monomorphisms
- has cartesian filtered colimits
- has coproducts
- is countably extensive
- is finitely cocomplete
- has connected colimits
- has coequalizers
- is multi-cocomplete
- has coquotients of cocongruences
- is Barr-coexact
- has finite products
- has a multi-terminal object
- has coreflexive equalizers
- is Cauchy complete
- has countable coproducts
- is extensive
- is filtered
- has directed colimits
- has ℵ₁-filtered colimits
- has sifted colimits
- has powers
- has ℵ₂-small products
- has wide pullbacks
- has finite coproducts
- has a multi-initial object
- has reflexive coequalizers
- is cofiltered
- has cosifted limits
- has copowers
- has ℵ₂-small coproducts
- has wide pushouts
- is mono-regular
- has disjoint finite coproducts
- has a strict initial object
- is distributive
- is infinitary distributive
- is countably distributive
- is sifted
- is ℵ₁-filtered
- has countable products
- has ℵ₂-small powers
- has binary products
- has a terminal object
- has finite powers
- has cofiltered limits
- has pullbacks
- is a pretopos
- is cosifted
- has sequential colimits
- has ℵ₂-small copowers
- has binary coproducts
- has an initial object
- has countable copowers
- has finite copowers
- has pushouts
- has a natural numbers object
- is connected
- has disjoint coproducts
- has cocartesian cofiltered limits
- has sequential limits
- is balanced
- has countable powers
- has binary powers
- is ℵ₁-cofiltered
- has directed limits
- has ℵ₁-cofiltered limits
- has binary copowers
- is epi-regular
- is inhabited
Unsatisfied Properties
Assigned properties
- is not skeletal
- is not Malcev
- is not semi-strongly connected
- is not locally essentially small
- is not cartesian closed
- is not well-powered
- does not have cofiltered-limit-stable epimorphisms
Deduced properties*
- is not thin
- is not additive
- is not accessible
- is not preadditive
- is not locally cartesian closed
- is not strongly connected
- is not discrete
- is not essentially discrete
- is not a groupoid
- is not essentially small
- is not locally small
- is not locally finite
- is not gaunt
- is not direct
- is not an elementary topos
- is not a Grothendieck topos
- is not coaccessible
- is not well-copowered
- is not coextensive
- does not have exact cofiltered limits
- is not right cancellative
- is not quotient-trivial
- is not essentially finite
- is not inverse
- is not self-dual
- is not locally presentable
- is not ℵ₁-accessible
- is not locally multi-presentable
- is not locally poly-presentable
- is not abelian
- is not finitary algebraic
- is not left cancellative
- does not have zero morphisms
- is not core-connected
- is not trivial
- does not have a strict terminal object
- is not small
- is not finite
- is not essentially countable
- is not regular-subobject-trivial
- is not core-thin
- is not a quasitopos
- is not locally copresentable
- is not cocartesian coclosed
- does not have disjoint finite products
- is not countably coextensive
- does not have a regular quotient object classifier
- is not regular-quotient-trivial
- is not pointed
- is not locally ℵ₁-presentable
- is not Grothendieck abelian
- is not finitely accessible
- is not split abelian
- does not have biproducts
- is not a generalized variety
- does not have kernels
- does not satisfy CIP
- is not normal
- is not countable
- is not subobject-trivial
- is not one-way
- is not locally cocartesian coclosed
- does not have disjoint products
- is not codistributive
- does not have cokernels
- does not satisfy CSP
- is not infinitary coextensive
- is not conormal
- does not have a quotient object classifier
- is not unital
- is not locally finitely presentable
- is not locally strongly finitely presentable
- is not locally finitely multi-presentable
- is not multi-algebraic
- is not counital
- is not countably codistributive
- is not infinitary codistributive
*This also uses the deduced satisfied properties.
Unknown properties
There are 10 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 an extremal cogenerating set
- has an extremal cogenerator
- has an extremal generating set
- has an extremal generator
- has a generating set
- has a generator
- has a regular subobject classifier
- has a subobject classifier
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