pushoutObjIso — Mathlib

English

In a pullback context for F,G,H and arrows f,g, the inverse of the pullback object iso composed with the first projection recovers the componentwise first projection at d.

Русский

В контексте pullback для F,G,H и стрелок f,g, обратная составная pullbackObjIso с fst восстанавливает fst на компоненте d.

LaTeX

$$$ (pullbackObjIso\\, f\\, g\\, d).inv \\;\\circ\\; (pullback.fst\\, f\\, g).app\\, d = pullback.fst\\left( f.{\\rm app}\\, d\\right)\\left( g.{\\rm app}\\, d\\right) $$$

Lean4

/-- Evaluating a pushout amounts to taking the pushout of the evaluations. -/
noncomputable def pushoutObjIso (f : F ⟶ G) (g : F ⟶ H) (d : D) : (pushout f g).obj d ≅ pushout (f.app d) (g.app d) :=
  colimitObjIsoColimitCompEvaluation (span f g) d ≪≫ HasColimit.isoOfNatIso (diagramIsoSpan _)

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