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
Properties from the database
- is coextensive
- is finitary algebraic
- is locally small
- is Malcev
- has a strict terminal object
Deduced properties
- is locally essentially small
- has a generator
- has a generating set
- is inhabited
- is locally strongly finitely presentable
- is well-copowered
- is regular
- is finitely complete
- has equalizers
- has finite products
- has binary products
- has a terminal object
- is connected
- has pullbacks
- is Cauchy complete
- is locally finitely presentable
- is locally presentable
- is locally ℵ₁-presentable
- is cocomplete
- is complete
- has cofiltered limits
- has connected limits
- has wide pullbacks
- has products
- has countable products
- has sequential limits
- is well-powered
- has exact filtered colimits
- has filtered colimits
- has directed colimits
- has connected colimits
- is finitely cocomplete
- has wide pushouts
- has sequential colimits
- has coequalizers
- has coproducts
- has countable coproducts
- has finite coproducts
- has binary coproducts
- has an initial object
- has pushouts
- has directed limits
- has disjoint finite products
- has disjoint products
- is codistributive
Unsatisfied Properties
Properties from the database
- is not balanced
- is not co-Malcev
- does not have a cogenerating set
- is not coregular
- is not infinitary codistributive
- is not skeletal
- does not have a strict initial object
- is not strongly connected
Deduced properties*
- is not left cancellative
- is not right cancellative
- is not distributive
- is not infinitary distributive
- is not cartesian closed
- is not extensive
- is not infinitary extensive
- is not lextensive
- is not a groupoid
- is not mono-regular
- 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 discrete
- is not essentially discrete
- is not trivial
- is not thin
- does not have disjoint finite coproducts
- does not have disjoint coproducts
- is not essentially small
- is not small
- is not finite
- is not essentially finite
- is not self-dual
- is not an elementary topos
- does not have a subobject classifier
- does not have a regular subobject classifier
- is not locally cartesian closed
- is not a Grothendieck topos
- is not unital
- is not infinitary coextensive
- does not have a cogenerator
- is not epi-regular
- is not counital
*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:
- 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.