nLab
simplicial homotopy equivalence

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

A simplicial homotopy equivalence is a homotopy equivalence between simplicial sets.

Definition

A morphism f:ABf\colon A\to B of simplicial sets is a simplicial homotopy equivalence if there is a morphism g:BAg\colon B\to A and homotopies p:Δ 1×AAp\colon \Delta^1\times A\to A from id Aid_A to gfg \circ f and q:Δ 1×BBq\colon \Delta^1\times B\to B from fgf \circ g to id Bid_B (where id Xid_X denotes the identity morphism on the simplicial set XX).

Properties

Last revised on October 6, 2020 at 02:10:48. See the history of this page for a list of all contributions to it.