projective_of_map_projective — Mathlib

English

If adj is an adjunction F ⊣ G with F full and faithful, and P is projective in C, then P is projective in C.

Русский

Если приложенный к F тождественный F ⊣ G, F полно и полноценно сохраняет морфизмы, и P проективен в C, тогда P проективен и в C.

LaTeX

$$$ (\\forall P)\\ (\\mathrm{Projective}(F\\!.(P)) \\Rightarrow \\mathrm{Projective}(P)) \\text{ for } F \\text{ full and faithful}. $$$

Lean4

theorem projective_of_map_projective (adj : F ⊣ G) [F.Full] [F.Faithful] (P : C) (hP : Projective (F.obj P)) :
    Projective P where
  factors f g
    _ := by
    haveI := Adjunction.leftAdjoint_preservesColimits.{0, 0} adj
    rcases (@hP).1 (F.map f) (F.map g) with ⟨f', hf'⟩
    use adj.unit.app _ ≫ G.map f' ≫ (inv <| adj.unit.app _)
    exact F.map_injective (by simpa)

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