instPreservesFiniteCoproductsToCompHausLike

English

The functor to TopCat preserves pullbacks of inclusions when inclusions exist.

Русский

Функтор в TopCat сохраняет пуллбэки включений, если они существуют.

LaTeX

$$$\\forall P'\\; (h:\\forall X, P X \\Rightarrow P' X)\\;\\Rightarrow\\text{PreservesPullbacksOfInclusions}$$$

Lean4

/-- The functor to another `CompHausLike` preserves finite coproducts if they exist. -/
noncomputable instance {P' : TopCat.{u} → Prop} (h : ∀ (X : CompHausLike P), P X.toTop → P' X.toTop) :
    PreservesFiniteCoproducts (toCompHausLike h) :=
  by
  have : PreservesFiniteCoproducts (toCompHausLike h ⋙ compHausLikeToTop P') :=
    inferInstanceAs (PreservesFiniteCoproducts (compHausLikeToTop _))
  exact preservesFiniteCoproducts_of_reflects_of_preserves (toCompHausLike h) (compHausLikeToTop P')

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