cokernelIsoRangeQuotient — Mathlib

English

The categorical cokernel of f is isomorphic to the quotient module H / range(f.hom).

Русский

Категориальное коканеллообразование f изоморфно к модулю-отношению H / range(f.hom).

LaTeX

$$$$ \\text{cokernel}(f) \\cong \\mathrm{ModuleCat.of}(R, H / \\operatorname{range}(f.hom)) $$$$

Lean4

/-- The categorical cokernel of a morphism in `ModuleCat`
agrees with the usual module-theoretical quotient.
-/
noncomputable def cokernelIsoRangeQuotient {G H : ModuleCat.{v} R} (f : G ⟶ H) :
    -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11036): broken dot notation
    cokernel f ≅ ModuleCat.of R (H ⧸ LinearMap.range f.hom) :=
  colimit.isoColimitCocone
    ⟨_, cokernelIsColimit f⟩
      -- We now show this isomorphism commutes with the projection of target to the cokernel.

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