ofArrows_pullback — Mathlib

English

The presieve built from pullbacks along g_i with f equals the pullback arrows of f with the presieve of arrows.

Русский

Предсистема, построенная из вытянутых диагоналей along g_i и f, равна стрелочным представлениям pullback.

LaTeX

$$$\\mathrm{ofArrows}\\ (\\lambda i.\\mathrm{pullback}(g_i,f))\\ (\\lambda i.\\mathrm{pullback}.snd\\; (g_i)\\, f) = \\mathrm{pullbackArrows}\\ f\\ (\\mathrm{ofArrows}\\ Z g)$$$

Lean4

theorem ofArrows_pullback {ι : Type*} (Z : ι → C) (g : ∀ i : ι, Z i ⟶ X) [∀ i, HasPullback (g i) f] :
    (ofArrows (fun i => pullback (g i) f) fun _ => pullback.snd _ _) = pullbackArrows f (ofArrows Z g) :=
  by
  funext T
  ext h
  constructor
  · rintro ⟨hk⟩
    exact pullbackArrows.mk _ _ (ofArrows.mk hk)
  · rintro ⟨W, k, ⟨_⟩⟩
    apply ofArrows.mk

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