category of algebras
- notation:
- objects: 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 rings, which we get for . 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
- has disjoint finite products
- 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 products
Unsatisfied Properties
Properties from the database
- is not balanced
- is not co-Malcev
- is not codistributive
- does not have a cogenerating set
- 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 codistributive
- is not coextensive
- 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
There is 1 property for which the database doesn't have an answer if it is satisfied or not. Please help to contribute the data!
- is coregular
Special objects
- terminal object: trivial algebra
- initial object:
- products: direct products with pointwise operations
- coproducts: see MSE/625874
Special morphisms
- isomorphisms: bijective ring homomorphisms
- monomorphisms: injective ring 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 algebras. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.