Simplicial Methods for Higher Categories by Simona Paoli

Author:Simona Paoli
Language: eng
Format: epub
ISBN: 9783030056742
Publisher: Springer International Publishing

Let 1 ≤ m < n. Then f : X → Y  in is a levelwise (n − m)-equivalence if and only if it is an n-equivalence and a) The map is an isomorphism.

b) For each 1 ≤ r < m and k 1, …, k r ∈ Δ, the maps

are isomorphisms.

Proof

Suppose that f is a levelwise (n − m)-equivalence. We proceed by induction on m. When m = 1, this is Lemma 7.1.3. Suppose, inductively, that the lemma holds for (m − 1) and let f : X → Y  be a levelwise (n − m)-equivalence. Then for each k 1 ≥ 0, is a levelwise (n − m + 1)-equivalence. So by the inductive hypothesis applied to , is an isomorphism, that is, b) holds.

When k 1 = 0, f 0 being a levelwise (n − m + 1)-equivalence implies by the inductive hypothesis that f 0 is an (n − 1)-equivalence in and thus by Lemma 5.​2.​6 pf 0 is an isomorphism, that is, a) holds.

It remains to prove that f is an n-equivalence. By the inductive hypothesis applied to f 1, f 1 is an (n − 1)-equivalence. Since

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