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
Assigned properties
- is locally small
- is split abelian
- is finitary algebraic
- has an extremal generator
- has an extremal cogenerator
Deduced properties
- is abelian
- is locally strongly finitely presentable
- is well-copowered
- has an extremal generating set
- has a generator
- is locally essentially small
- has a cogenerator
- has an extremal cogenerating set
- is locally finitely presentable
- is cocomplete
- is a generalized variety
- is additive
- has cokernels
- is conormal
- has kernels
- is normal
- is regular
- is multi-algebraic
- has a generating set
- is inhabited
- is coregular
- has a cogenerating set
- is finitely accessible
- has exact filtered colimits
- is locally ℵ₁-presentable
- has finite products
- is preadditive
- has biproducts
- has sifted colimits
- is ℵ₁-accessible
- has filtered-colimit-stable monomorphisms
- is locally finitely multi-presentable
- is multi-cocomplete
- has effective congruences
- is finitely complete
- has zero morphisms
- is mono-regular
- has finite coproducts
- has connected colimits
- is finitely cocomplete
- has coequalizers
- has coproducts
- is epi-regular
- is Malcev
- is unital
- is locally presentable
- is accessible
- has ℵ₁-filtered colimits
- has filtered colimits
- has connected limits
- is Grothendieck abelian
- has equalizers
- has quotients of congruences
- is Barr-exact
- is strongly connected
- has cartesian filtered colimits
- is filtered
- has reflexive coequalizers
- is balanced
- has binary products
- has a terminal object
- has finite powers
- is co-Malcev
- is counital
- has a multi-initial object
- is well-powered
- has coquotients of cocongruences
- has effective cocongruences
- is Cauchy complete
- is cofiltered
- has copowers
- has ℵ₂-small coproducts
- has binary coproducts
- has an initial object
- has finite copowers
- has wide pushouts
- is pointed
- is complete
- is locally multi-presentable
- is connected
- has a multi-terminal object
- is semi-strongly connected
- has disjoint finite products
- has coreflexive equalizers
- is sifted
- is ℵ₁-filtered
- has directed colimits
- has binary powers
- has pullbacks
- has wide pullbacks
- is Barr-coexact
- has disjoint finite coproducts
- is cosifted
- is ℵ₁-cofiltered
- has cosifted limits
- has countable coproducts
- has ℵ₂-small copowers
- has binary copowers
- has pushouts
- is locally poly-presentable
- has products
- is multi-complete
- has disjoint coproducts
- has cofiltered limits
- has sequential colimits
- has countable copowers
- satisfies CIP
- has powers
- has ℵ₂-small products
- has disjoint products
- has cocartesian cofiltered limits
- has directed limits
- has ℵ₁-cofiltered limits
- has sequential limits
- has countable products
- has ℵ₂-small powers
- has countable powers
Unsatisfied Properties
Assigned properties
Deduced properties*
- is not discrete
- is not gaunt
- is not direct
- does not have cofiltered-limit-stable epimorphisms
- is not inverse
- is not self-dual
- is not trivial
- is not a groupoid
- does not have exact cofiltered limits
- is not right cancellative
- is not quotient-trivial
- is not essentially finite
- does not have a natural numbers object
- is not thin
- is not core-connected
- is not essentially discrete
- does not have a strict initial object
- is not finite
- does not have a regular subobject classifier
- is not regular-subobject-trivial
- is not regular-quotient-trivial
- does not have a strict terminal object
- does not have a regular quotient object classifier
- is not left cancellative
- is not countably distributive
- is not cartesian closed
- is not distributive
- is not extensive
- does not have a subobject classifier
- is not subobject-trivial
- is not core-thin
- is not locally finite
- is not one-way
- is not essentially small
- is not essentially countable
- is not a quasitopos
- is not cocartesian coclosed
- is not codistributive
- is not coextensive
- does not have a quotient object classifier
- is not locally copresentable
- is not locally cartesian closed
- is not infinitary distributive
- is not countably extensive
- is not small
- is not countable
- is not an elementary topos
- is not a pretopos
- is not locally cocartesian coclosed
- is not countably codistributive
- is not countably coextensive
- is not infinitary extensive
- is not a Grothendieck topos
- is not coaccessible
- is not infinitary codistributive
- is not infinitary coextensive
*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 morphisms
- monomorphisms: injective morphisms
- epimorphisms: surjective morphisms
- regular monomorphisms: same as monomorphisms
- regular epimorphisms: surjective morphisms
Indistinguishable 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.
Functors
The database has stored 1 functor whose (co-)domain is the category of vector spaces.