componentHom — Mathlib

English

The counit is natural in S and Y: a natural transformation from functorToPresheaves.obj (Y.obj ⟨of P PUnit⟩) to Y, whose component at S is counitApp S Y.

Русский

Каутит — естественный по отношению к S и Y: естественная трансформация от functorToPresheaves.obj (Y.obj ⟨of P PUnit⟩) к Y, компонентом в S является counitApp S Y.

LaTeX

$$$\text{counitApp} : (\text{functorToPresheaves.obj}(Y.obj⟨P\;PUnit\rangle)) \Rightarrow Y$$$

Lean4

/-- This is an auxiliary definition, the details do not matter. What's important is that this map exists
so that the lemma `incl_comap` works.
-/
def componentHom (a : Fiber (f.comap g.hom)) : fiber _ a ⟶ fiber _ (Fiber.mk f (g a.preimage)) :=
  TopCat.ofHom
    { toFun
        x :=
        ⟨g x.val, by
          simp only [Fiber.mk, Set.mem_preimage, Set.mem_singleton_iff]
          convert map_eq_image _ _ x
          exact map_preimage_eq_image_map _ _ a⟩
      continuous_toFun := by exact Continuous.subtype_mk (g.hom.continuous.comp continuous_subtype_val) _ }
      -- term mode gives "unknown free variable" error.

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