category of large families of abelian groups

 algebra
Notation AbI\Ab^I Objects families of abelian groups (Ai)i∈I(A_i)_{i \in I} indexed by a collection II that is not essentially small Morphisms families of homomorphisms Related Ab\Ab, grAb\grAb, [On,Ab][\On,\Ab], VectK(I)\Vect^{(I)}_K, SetI\Set^I

This is the product category AbI=∏i∈IAb\Ab^I = \prod_{i \in I} \Ab, or equivalently, the functor category [Idisc⁡,Ab][I_{\disc},\Ab]. For some background on why this product category is well-defined even though II is a collection, see Foundations. This category is a larger variant of grAb=AbZ\grAb = \Ab^{\IZ}. The properties do not depend on the specific choice of II, but to make things concrete, one might take I=SetI = \SetColl, the collection of all sets. The category does not appear in practice, but we have added it because of its interesting combinations of properties. For example, it shows that a Grothendieck abelian category is not necessarily locally essentially small or well-powered.

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 groups
  • initial object: family of trivial groups
  • products: pointwise defined direct products
  • coproducts: pointwise defined direct sums

Special morphisms

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