nLab locally contractible topological infinity-groupoid

Context

Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Structures in a cohesive $(\infty,1)$-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Topology

topology

algebraic topology

Contents

Idea

A locally contractible topological ∞-groupoid is an ∞-groupoid equipped with cohesion in the form of locally contractible topology.

The collection of all these cohesive $\infty$-groupoids forms a cohesive (∞,1)-topos $LCTop\infty Grpd$.

This is similar to ETop∞Grpd, which models cohesion in the form of Euclidean topology.

Definition

Let $CTop$ be some small version (…details missing…) of the site of locally contractible contractible topological spaces with continuous maps betwen them and equipped with the standard open cover coverage.

This is a cohesive site (for the evident generalization of that definitions where Cech covers are generalized to hypercovers). The key axiom to check is that for $Y \to U$ a hypercover of $U \in CTop$ degreewise by a coproduct of contractibles, also the simplicial set $\lim_\to Y$ obtained by sending each contractible to a point is contractible. This follows as pointed out on MO here.1

Define then

$LCTop\infty Grpd := Sh_{(\infty,1)}(CTop)$

to be the (∞,1)-category of (∞,1)-sheaves on $CTop$.

This is an cohesive (∞,1)-topos.

$(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : LCTop\infty Grpd \to \infty Grpd \,.$

1. Thanks to David Carchedi for highlighting this.

Revised on January 7, 2015 10:31:47 by Tim Porter (127.0.0.1)