category of long transfinite sequences of abelian groups

 algebra
Notation [On,Ab][\On,\Ab] Objects long transfinite sequences of abelian groups, i.e. families of abelian groups (Aα)(A_\alpha) indexed by all ordinal numbers, equipped with transition homomorphisms Aα,β:Aα→AβA_{\alpha,\beta} : A_\alpha \to A_\beta for α≤β\alpha \leq \beta satisfying Aα,α=id⁡AαA_{\alpha,\alpha} = \id_{A_\alpha} and Aα,γ=Aβ,γ∘Aα,βA_{\alpha,\gamma} = A_{\beta,\gamma} \circ A_{\alpha,\beta} for α≤β≤γ\alpha \leq \beta \leq \gamma Morphisms A morphism A→BA \to B is a family of homomorphisms Aα→BαA_\alpha \to B_\alpha such that the evident square commutes for every α≤β\alpha \leq \beta. Related Ab\Ab, VectKs\Vect^s_K, Ab(N,≤)\Ab^{(\IN,\leq)}, [Setdisc⁡,Ab][\Set_{\disc},\Ab], On\On

This is the functor category [On,Ab][\On,\Ab], where On\On is the thin category of ordinal numbers. It is a larger variant of the category of sequences of abelian groups [N,Ab][\IN,\Ab]. This category rarely appears in practice, but we have added it because of its interesting combinations of properties. For example, it shows that a Grothendieck abelian category that is not locally small does not necessarily have a cogenerator. 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: sequence of trivial groups
  • initial object: sequence of trivial groups
  • products: pointwise defined direct products
  • coproducts: pointwise defined direct sums

Special morphisms

  • isomorphisms: morphisms that are pointwise bijective
  • monomorphisms: morphisms that are pointwise injective
  • epimorphisms: morphisms that are pointwise surjective
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: same as epimorphisms