category of set-indexed families of abelian groups

 algebra
Notation [Setdisc⁡,Ab][\Set_{\disc},\Ab] Objects families of abelian groups (AX)X∈Set(A_X)_{X \in \SetColl} indexed by all sets Morphisms families of homomorphisms Related Ab\Ab, grAb\grAb, VectKs\Vect^s_K, [On,Ab][\On,\Ab], [CRing,Set][\CRing, \Set], Setdisc⁡\Set_{\disc}

This functor category [Setdisc⁡,Ab]≅AbSet[\Set_{\disc},\Ab] \cong \Ab^{\SetColl} is a larger variant of grAb=[Zdisc⁡,Ab]\grAb = [\IZ_{\disc}, \Ab]. Instead of Setdisc⁡\Set_{\disc}, we may take any other large discrete category. It 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 small. For some background on why this functor category is well-defined, 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 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