category of large families of vector spaces which are mostly zero

 algebra
Notation VectK(I)\Vect^{(I)}_K Objects families of vector spaces V=(Vi)i∈IV = (V_i)_{i \in I} over a field KK whose support supp⁡(V)≔{i∈I:Vi≠0}\supp(V) \coloneqq \{i \in I : V_i \neq 0\} is essentially small (i.e., isomorphic to a set), where II is a fixed collection that is not essentially small Morphisms families of linear maps Related VectK\Vect_K, AbI\Ab^I, VectKs\Vect^s_K, Set0(I)\Set^{(I)}_0, Set1(I)\Set^{(I)}_1, SetI\Set^I

We have added this category solely as an example of a split abelian category that does not have a generator. It is a full subcategory of the product category VectKI\Vect_K^I (which exists even when II is merely a collection; see Foundations).

Satisfied Properties

Assigned properties

Deduced properties

Unsatisfied Properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: family of trivial vector spaces
  • initial object: family of trivial vector spaces
  • products: pointwise direct products
  • coproducts: pointwise direct sums

Special morphisms

  • isomorphisms: families of bijective linear maps
  • monomorphisms: families of injective linear maps
  • epimorphisms: families of surjective linear maps
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: same as epimorphisms