category of vector spaces
- notation:
- objects: vector spaces over a field
- morphisms: linear maps
- Related categories: ,
- nLab Link
This is a special case of the category of modules over a ring, where the ring is a field. It is the prototype of a split abelian category.
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
- has finite powers
- has binary powers
- is preadditive
- has zero morphisms
- is strongly connected
- has biproducts
- has finite coproducts
- has disjoint finite coproducts
- has coequalizers
- is epi-regular
- has equalizers
- has coreflexive equalizers
- is finitely complete
- has pullbacks
- is Cauchy complete
- is unital
- 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 connected limits
- has products
- has countable products
- has wide pullbacks
- has cofiltered limits
- has sequential limits
- has powers
- has countable powers
- is well-powered
- has exact filtered colimits
- has filtered colimits
- has directed colimits
- is Malcev
- is pointed
- has an initial object
- has disjoint products
- has connected colimits
- has sifted colimits
- has reflexive coequalizers
- is finitely cocomplete
- has cosifted limits
- has binary coproducts
- has coproducts
- has disjoint coproducts
- is Grothendieck abelian
- has a cogenerator
- has countable coproducts
- has sequential colimits
- has pushouts
- has directed limits
- has wide pushouts
- has copowers
- has countable copowers
- has finite copowers
- has binary copowers
- 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 direct
- is not one-way
- is not discrete
- is not self-dual
- 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
- is not inverse
*This also uses the deduced satisfied properties.
Unknown properties
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Special objects
- terminal object: trivial vector space
- initial object: trivial vector space
- products: direct products with pointwise operations
- coproducts: direct sums
Special morphisms
- isomorphisms: bijective linear maps
- monomorphisms: injective linear maps
- epimorphisms: surjective linear maps
- 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 vector spaces. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.