fiberEqualizerEquiv — Mathlib

English

The fiber of the equalizer is equivalent to the agreement set of f and g.

Русский

Фибер равнозазначающего эквивалентен множеству согласований условий $F(f)$ и $F(g)$.

LaTeX

$$$\\mathrm{F.obj}(\\mathrm{equalizer}(f,g)) \\simeq \\{ x:\\, \\mathrm{F.obj}(X) \\;|\\; \\mathrm{F.map}(f)x = \\mathrm{F.map}(g)x \\}$$$

Lean4

/-- The fiber of the equalizer of `f g : X ⟶ Y` is equivalent to the set of agreement of `f`
and `g`. -/
noncomputable def fiberEqualizerEquiv {X Y : C} (f g : X ⟶ Y) :
    F.obj (equalizer f g) ≃ { x : F.obj X // F.map f x = F.map g x } :=
  (PreservesEqualizer.iso (F ⋙ FintypeCat.incl) f g ≪≫ Types.equalizerIso (F.map f) (F.map g)).toEquiv

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