category of finite-dimensional vector spaces
- Notation:
- Objects: finite-dimensional vector spaces over a field
- Morphisms: linear maps
- Children: , ,
- Related categories: ,
- nLab Link
This is the full subcategory of consisting of finite-dimensional vector spaces. Every object is isomorphic to for a unique .
Satisfied Properties
Assigned properties
- is locally small
- is essentially small
- has an extremal generator
- is split abelian
- is self-dual
- is ℵ₁-accessible
Deduced properties
- is accessible
- has ℵ₁-filtered colimits
- is abelian
- has an extremal generating set
- has a generator
- is locally essentially small
- is well-copowered
- is well-powered
- has an extremal cogenerating set
- has an extremal cogenerator
- is Cauchy complete
- is additive
- has cokernels
- is conormal
- has kernels
- is normal
- is regular
- has a generating set
- is inhabited
- is coregular
- has a cogenerator
- has a cogenerating set
- is coaccessible
- has ℵ₁-cofiltered limits
- has finite products
- is preadditive
- has biproducts
- is finitely complete
- has zero morphisms
- is mono-regular
- has finite coproducts
- is finitely cocomplete
- is epi-regular
- is Malcev
- is unital
- has equalizers
- has quotients of congruences
- has effective congruences
- is strongly connected
- is filtered
- is balanced
- has binary products
- has a terminal object
- has finite powers
- is co-Malcev
- is counital
- has coequalizers
- has coquotients of cocongruences
- has effective cocongruences
- is cofiltered
- has binary coproducts
- has an initial object
- has finite copowers
- is pointed
- is connected
- has a multi-terminal object
- is Barr-exact
- is semi-strongly connected
- has disjoint finite products
- has coreflexive equalizers
- is sifted
- is ℵ₁-filtered
- has binary powers
- has pullbacks
- has a multi-initial object
- is Barr-coexact
- has disjoint finite coproducts
- has reflexive coequalizers
- is cosifted
- is ℵ₁-cofiltered
- has binary copowers
- has pushouts
- has a natural numbers object
Unsatisfied Properties
Assigned properties
Deduced properties*
- is not left cancellative
- is not discrete
- is not essentially discrete
- does not have a strict terminal object
- is not a groupoid
- is not finite
- is not regular-subobject-trivial
- is not core-thin
- is not gaunt
- is not direct
- is not one-way
- does not have powers
- is not essentially finite
- is not right cancellative
- does not have a strict initial object
- is not regular-quotient-trivial
- is not inverse
- does not have copowers
- is not cartesian closed
- is not trivial
- is not distributive
- is not extensive
- does not have products
- does not have filtered colimits
- does not have countable copowers
- is not subobject-trivial
- does not have a regular subobject classifier
- is not cocartesian coclosed
- is not codistributive
- is not coextensive
- does not have coproducts
- does not have cofiltered limits
- does not have countable powers
- is not quotient-trivial
- does not have a regular quotient object classifier
- does not have a parametrized natural numbers object
- is not finitely accessible
- is not Grothendieck abelian
- is not locally cartesian closed
- is not complete
- is not core-connected
- does not have disjoint coproducts
- is not infinitary distributive
- is not countably distributive
- does not have exact filtered colimits
- does not have cartesian filtered colimits
- does not have filtered-colimit-stable monomorphisms
- does not satisfy CIP
- is not infinitary extensive
- is not countably extensive
- does not have directed colimits
- does not have sifted colimits
- does not have countable products
- does not have ℵ₂-small powers
- does not have sequential limits
- does not have wide pullbacks
- does not have a subobject classifier
- is not an elementary topos
- is not a Grothendieck topos
- is not a pretopos
- is not a quasitopos
- is not locally cocartesian coclosed
- is not cocomplete
- does not have disjoint products
- is not infinitary codistributive
- is not countably codistributive
- does not have exact cofiltered limits
- does not have cocartesian cofiltered limits
- does not have cofiltered-limit-stable epimorphisms
- does not satisfy CSP
- is not infinitary coextensive
- is not countably coextensive
- does not have directed limits
- does not have cosifted limits
- does not have countable coproducts
- does not have ℵ₂-small copowers
- does not have sequential colimits
- does not have wide pushouts
- does not have a quotient object classifier
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not locally presentable
- is not finitary algebraic
- is not locally finitely multi-presentable
- is not locally poly-presentable
- is not a generalized variety
- is not multi-complete
- does not have connected colimits
- does not have ℵ₂-small products
- does not have connected limits
- is not locally copresentable
- is not multi-cocomplete
- does not have ℵ₂-small coproducts
- is not locally multi-presentable
- is not one-sorted finitary algebraic
- is not multi-algebraic
*This also uses the deduced satisfied properties.
Unknown properties
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Undecidable properties
There are 2 properties for which it cannot be decided if they are satisfied or not.
Special objects
- terminal object: the trivial vector space
- initial object: the trivial vector space
- products: [finite case] direct products with pointwise operations
- coproducts: [finite case] direct sums
Special morphisms
- isomorphisms: bijective linear maps
- monomorphisms: injective linear maps
- epimorphisms: surjective linear maps
- regular monomorphisms: same as monomorphisms
- regular epimorphisms: same as epimorphisms
Symmetric monoidal categories
The database stores 1 symmetric monoidal category based on the category of finite-dimensional vector spaces.