hasSmallLocalizedHom_iff_of_isos — Mathlib

English

If there is a localization L, HasSmallLocalizedHom for W between X and Y is equivalent to Small Hom in the localized category.

Русский

Если есть локализация L, то HasSmallLocalizedHom для W между X и Y эквивалентен SmallHom в локализованной категории.

LaTeX

$$$\text{HasSmallLocalizedHom}(W,X,Y) \iff \text{Small}(L.obj X \to L.obj Y).$$$

Lean4

theorem hasSmallLocalizedHom_iff_of_isos {X' Y' : C} (e : X ≅ X') (e' : Y ≅ Y') :
    HasSmallLocalizedHom.{w} W X Y ↔ HasSmallLocalizedHom.{w} W X' Y' :=
  by
  simp only [hasSmallLocalizedHom_iff W W.Q]
  exact small_congr (Iso.homCongr (W.Q.mapIso e) (W.Q.mapIso e'))

Похожие утверждения