category of commutative algebras
- notation:
- objects: commutative algebras over a commutative ring
- morphisms: maps preserving the ring and module structure
- Related categories: , ,
- nLab Link
This is a generalization of the category of commutative rings, which we get for . In general, . We assume our rings (and algebras) to be unital. For we would get the trivial category, which is why we exclude this here.
Satisfied Properties
Assigned properties
- is locally small
- is finitary algebraic
- has a strict terminal object
- is Malcev
- is coextensive
Deduced properties
- is finitely complete
- is locally strongly finitely presentable
- is well-copowered
- has a generator
- is locally essentially small
- has finite products
- has disjoint finite products
- has a terminal object
- is locally finitely presentable
- is cocomplete
- is a generalized variety
- is regular
- is multi-algebraic
- has equalizers
- is connected
- has a multi-terminal object
- is filtered
- has a generating set
- is inhabited
- has binary products
- has finite powers
- is cofiltered
- is finitely accessible
- has exact filtered colimits
- is locally ℵ₁-presentable
- has sifted colimits
- is ℵ₁-accessible
- is locally finitely multi-presentable
- is multi-cocomplete
- has effective congruences
- has coreflexive equalizers
- is Cauchy complete
- is sifted
- has binary powers
- has pullbacks
- has connected colimits
- is finitely cocomplete
- has coequalizers
- has coproducts
- is cosifted
- is locally presentable
- is accessible
- has filtered colimits
- has connected limits
- has filtered-colimit-stable monomorphisms
- has quotients of congruences
- is Barr-exact
- has cartesian filtered colimits
- has reflexive coequalizers
- has finite coproducts
- has a multi-initial object
- has coquotients of cocongruences
- has copowers
- has countable coproducts
- has wide pushouts
- is well-powered
- is complete
- is locally multi-presentable
- has directed colimits
- has wide pullbacks
- has an initial object
- is codistributive
- has sequential colimits
- has cosifted limits
- has binary coproducts
- has countable copowers
- has finite copowers
- has pushouts
- is locally poly-presentable
- has products
- is multi-complete
- has cofiltered limits
- has binary copowers
- has countable products
- has powers
- has disjoint products
- has directed limits
- has sequential limits
- has countable powers
Unsatisfied Properties
Assigned properties
- is not skeletal
- is not balanced
- does not have a cogenerating set
- is not countably codistributive
- is not semi-strongly connected
- is not coregular
- is not co-Malcev
- does not have a regular quotient object classifier
- does not have cofiltered-limit-stable epimorphisms
Deduced properties*
- is not strongly connected
- is not discrete
- is not mono-regular
- is not gaunt
- is not direct
- is not an elementary topos
- is not thin
- is not additive
- is not coaccessible
- is not abelian
- is not locally cocartesian coclosed
- is not Barr-coexact
- is not infinitary codistributive
- is not cocartesian coclosed
- does not have cocartesian cofiltered limits
- does not have exact cofiltered limits
- is not right cancellative
- is not quotient-trivial
- is not essentially finite
- does not have a cogenerator
- is not epi-regular
- is not essentially small
- is not left cancellative
- does not have a quotient object classifier
- is not inverse
- is not self-dual
- is not locally copresentable
- is not preadditive
- is not Grothendieck abelian
- is not split abelian
- is not cartesian closed
- is not extensive
- does not have zero morphisms
- is not trivial
- is not essentially discrete
- does not have disjoint finite coproducts
- is not a groupoid
- is not normal
- is not small
- is not finite
- is not essentially countable
- does not have a subobject classifier
- does not have a regular subobject classifier
- is not subobject-trivial
- is not core-thin
- is not locally finite
- is not a Grothendieck topos
- does not have effective cocongruences
- does not have a strict initial object
- does not have biproducts
- is not infinitary coextensive
- is not conormal
- is not locally cartesian closed
- does not have disjoint coproducts
- is not distributive
- does not have kernels
- does not satisfy CIP
- is not infinitary extensive
- is not pointed
- is not countable
- is not one-way
- does not have cokernels
- does not satisfy CSP
- is not unital
- does not have a natural numbers object
- is not countably distributive
- is not counital
- is not infinitary distributive
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- terminal object: trivial algebra
- initial object:
- products: direct products with pointwise operations
- coproducts: tensor products over
Special morphisms
- isomorphisms: bijective homomorphisms
- monomorphisms: injective homomorphisms
- epimorphisms: a homomorphism of algebras which is an epimorphism of commutative rings
- regular monomorphisms:
- regular epimorphisms: surjective homomorphisms
Undistinguishable categories
These categories in the database currently have exactly the same properties as the category of commutative algebras. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.