Theorems · Theorem · category theory

Function.Exact.iff_rangeFactorization

∀ {M : Type u_2} {N : Type u_4} {P : Type u_6} {f : M → N} {g : N → P} [inst : Zero P] (hg : 0 ∈ Set.range g),
  Function.Exact f g ↔ Function.Exact Subtype.val (Set.rangeFactorization g)

Two maps f : M → N and g : N → P are exact if and only if the induced maps Set.range f → N → Set.range g are exact. Note that if you already have an instance [Zero (Set.range g)] (which is unlikely) this lemma may not apply if the zero of Set.range g is not definitionally equal to ⟨0, hg⟩.

Defined in

Mathlib.Algebra.Exact.Basic

Cited by

3 results in Mathlib

Foundations

Depth 19 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

Zero