epi_of_nonempty_of_isConnected — Mathlib

English

If A is connected and F.obj X is nonempty, then every morphism f: X ⟶ A is an epi.

Русский

Если A связный и F.obj X непусто, то любой морфизм f: X ⟶ A является эпиморфизмом.

LaTeX

$$$\bigl(IsConnected\ A \land F.obj X \neq \varnothing\bigr) \Rightarrow \mathrm{Epi}(f)$$$

Lean4

/-- A morphism from an object `X` with non-empty fiber to a connected object `A` is an
epimorphism. -/
theorem epi_of_nonempty_of_isConnected {X A : C} [IsConnected A] [h : Nonempty (F.obj X)] (f : X ⟶ A) : Epi f :=
  Epi.mk <| fun {Z} u v huv ↦
    by
    apply evaluation_injective_of_isConnected F A Z (F.map f (Classical.arbitrary _))
    simpa using congr_fun (F.congr_map huv) _

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