underPost — Mathlib

English

The cokernel of a zero map is preserved by any functor that preserves zero morphisms; equivalently, PreservesColimit( parallelPair f 0 ) G when f = 0.

Русский

Кокernel нулевого отображения сохраняется любым функтором, сохраняющим нулевые морфизмы; эквивалентно PreservesColimit( parallelPair f 0 ) G при f = 0.

LaTeX

$$$f = 0 \Rightarrow \text{PreservesColimit}(\text{parallelPair } f\ 0)\ G$$$

Lean4

instance underPost [PreservesColimitsOfShape (WithInitial J) F] : PreservesColimitsOfShape J (Under.post F (X := X))
    where
  preservesColimit.preserves {coconeK}
    isColimitCoconeK :=
    have isColimitCoconeD :=
      (IsColimit.precomposeHomEquiv liftFromUnderComp _).symm <|
        isColimitOfPreserves F (isColimitEquiv.symm isColimitCoconeK)
    ⟨isColimitEquiv <|
        isColimitCoconeD.ofIsoColimit <|
          Cocones.ext (.refl _) fun
            | .star | .of a => by aesop⟩

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