CategoryTheory.ShortComplex.moduleCatCyclesIso_hom_subtype

theorem CategoryTheory.ShortComplex.moduleCatCyclesIso_hom_subtype : ∀ {R : Type u} [inst : Ring R] (S : CategoryTheory.ShortComplex (ModuleCat R)), CategoryTheory.CategoryStruct.comp S.moduleCatCyclesIso.hom (LinearMap.ker S.g).subtype = S.iCycles

The theorem `CategoryTheory.ShortComplex.moduleCatCyclesIso_hom_subtype` states that for any given type `R` equipped with a ring structure and for any short complex `S` over the category of `R`-modules, the composition of the morphism corresponding to the isomorphism between the abstract `S.cycles` of the homology API and the more concrete description as the kernel of `S.g`, with the embedding of the submodule corresponding to the kernel of `S.g` into the ambient space, is equal to the inclusion morphism from `S.cycles` to `S.X₂` in the short complex `S`. This theorem is about the relationship between different ways of expressing cycles in a short complex of modules in category theory.

More concisely: The isomorphism between the cycles of a short complex of R-modules and the kernel of its differential, followed by the embedding of the cycle submodule into the module of degree 2, is equal to the inclusion map.