category of left modules over a division ring
- notation:
- objects: left -modules
- morphisms: -linear maps
- Related categories: ,
- nLab Link
Here, we assume that is a non-commutative division ring, i.e. a skew-field which is not a field. The category of modules behaves mostly the same as in the commutative case.
Satisfied Properties
Properties from the database
- is finitary algebraic
- is locally small
- is split abelian
Deduced properties
- is locally essentially small
- has a generator
- has a generating set
- is inhabited
- is locally strongly finitely presentable
- is abelian
- is additive
- has finite products
- has binary products
- has a terminal object
- is connected
- is preadditive
- has zero morphisms
- is strongly connected
- has biproducts
- has finite coproducts
- has disjoint finite coproducts
- has coequalizers
- is epi-regular
- has equalizers
- is finitely complete
- has pullbacks
- is Cauchy complete
- is mono-regular
- is balanced
- is regular
- is well-copowered
- 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
- is Malcev
- has connected colimits
- is finitely cocomplete
- has wide pushouts
- has binary coproducts
- has an initial object
- is pointed
- is unital
- has disjoint products
- has coproducts
- has disjoint coproducts
- is Grothendieck abelian
- has a cogenerator
- has countable coproducts
- has sequential colimits
- has pushouts
- has directed limits
- is counital
- has disjoint finite products
- has a cogenerating set
- is coregular
- is co-Malcev
Unsatisfied Properties
Properties from the database
Deduced properties*
- is not thin
- does not have a strict terminal object
- does not have a strict initial object
- 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 self-dual
- is not discrete
- is not essentially discrete
- is not essentially small
- is not small
- is not finite
- is not essentially finite
- is not an elementary topos
- does not have a regular subobject classifier
- does not have a subobject classifier
- is not locally cartesian closed
- is not a Grothendieck topos
- is not codistributive
- is not infinitary codistributive
- is not coextensive
- is not infinitary coextensive
*This also uses the deduced satisfied properties.
Unknown properties
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Special objects
- terminal object: zero module
- initial object: trivial module
- products: direct products with pointwise operations
- coproducts: direct sums
Special morphisms
- isomorphisms:
- monomorphisms:
- epimorphisms:
- regular monomorphisms: same as monomorphisms
- regular epimorphisms: surjective homomorphisms
Undistinguishable categories
These categories in the database currently have exactly the same properties as the category of left modules over a division ring. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.