Theorems · Theorem · algebraic topology

CategoryTheory.SimplicialObject.Splitting.cofan_inj_comp_PInfty_eq_zero

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
  {X : CategoryTheory.SimplicialObject C} (s : X.Splitting) {n : ℕ}
  (A : CategoryTheory.SimplicialObject.Splitting.IndexSet (Opposite.op { len := n })),
  ¬A.EqId →
    CategoryTheory.CategoryStruct.comp ((s.cofan (Opposite.op { len := n })).inj A)
        (AlgebraicTopology.DoldKan.PInfty.f n) =
      0

If a simplicial object X in an additive category is split, then PInfty vanishes on all the summands of X _⦋n⦌ which do not correspond to the identity of ⦋n⦌.

Defined in

Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Cited by

2 results in Mathlib

Foundations

Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Preadditive