category of commutative algebras
- Notation:
- Objects: commutative algebras over a commutative ring
- Morphisms: maps preserving the ring and module structure
- Children:
- Related categories: ,
- nLab Link
This category 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 one-sorted finitary algebraic
- has a strict terminal object
- is Malcev
- is coextensive
Deduced properties
- is finitely complete
- has a natural numbers object
- is finitary algebraic
- is well-copowered
- has an extremal 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 ℵ₁-filtered
- has an extremal generating set
- has a generator
- 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
- has filtered-colimit-stable monomorphisms
- is locally finitely multi-presentable
- is multi-cocomplete
- has effective congruences
- is inhabited
- has coreflexive equalizers
- is Cauchy complete
- is filtered
- has a generating set
- has binary powers
- has pullbacks
- has connected colimits
- is finitely cocomplete
- has coequalizers
- has coproducts
- is cosifted
- is locally presentable
- is accessible
- has ℵ₁-filtered colimits
- has filtered colimits
- has connected limits
- has quotients of congruences
- is Barr-exact
- has cartesian filtered colimits
- is sifted
- has reflexive coequalizers
- has finite coproducts
- has a multi-initial object
- has coquotients of cocongruences
- has copowers
- has ℵ₂-small 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 cosifted limits
- has countable coproducts
- has ℵ₂-small copowers
- has binary coproducts
- has finite copowers
- has pushouts
- is locally poly-presentable
- has products
- is multi-complete
- has cofiltered limits
- is ℵ₁-cofiltered
- has sequential colimits
- has countable copowers
- has binary copowers
- has powers
- has ℵ₂-small products
- has disjoint products
- has directed limits
- has ℵ₁-cofiltered limits
- has sequential limits
- has countable products
- has ℵ₂-small powers
- 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 a quasitopos
- is not thin
- is not additive
- 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 countably coextensive
- is not essentially finite
- does not have a cogenerator
- does not have an extremal cogenerating set
- is not epi-regular
- is not left cancellative
- does not have a quotient object classifier
- is not inverse
- is not self-dual
- 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 core-connected
- is not trivial
- is not essentially discrete
- does not have disjoint finite coproducts
- is not a groupoid
- is not normal
- is not finite
- does not have a subobject classifier
- does not have a regular subobject classifier
- is not regular-subobject-trivial
- is not subobject-trivial
- is not core-thin
- is not locally finite
- is not essentially small
- is not essentially countable
- is not an elementary topos
- is not a Grothendieck topos
- is not coaccessible
- does not have effective cocongruences
- does not have a strict initial object
- does not have biproducts
- is not infinitary coextensive
- does not have an extremal cogenerator
- is not conormal
- is not regular-quotient-trivial
- is not locally copresentable
- is not locally cartesian closed
- does not have disjoint coproducts
- is not distributive
- does not have kernels
- does not satisfy CIP
- is not countably extensive
- is not pointed
- is not small
- is not countable
- is not one-way
- is not a pretopos
- does not have cokernels
- does not satisfy CSP
- is not unital
- does not have a parametrized natural numbers object
- is not countably distributive
- is not infinitary extensive
- 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 morphisms
- monomorphisms: injective morphisms
- epimorphisms: a homomorphism of algebras which is an epimorphism of commutative rings
- regular monomorphisms:
- regular epimorphisms: surjective morphisms
Indistinguishable 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.
Functors
The database stores 1 functor based on the category of commutative algebras.