mathlib3 documentation


Behaviour of P_infty with respect to degeneracies #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

For any X : simplicial_object C where C is an abelian category, the projector P_infty : K[X] ⟶ K[X] is supposed to be the projection on the normalized subcomplex, parallel to the degenerate subcomplex, i.e. the subcomplex generated by the images of all X.σ i.

In this file, we obtain degeneracy_comp_P_infty which states that if X : simplicial_object C with C a preadditive category, θ : [n] ⟶ Δ' is a non injective map in simplex_category, then θ.op ≫ P_infty.f n = 0. It follows from the more precise statement vanishing statement σ_comp_P_eq_zero for the P q.

(See equivalence.lean for the general strategy of proof of the Dold-Kan equivalence.)