nLab locally contractible topological infinity-groupoid

Context

Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Structures in a cohesive $\left(\infty ,1\right)$-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 $\mathrm{LCTop}\infty \mathrm{Grpd}$.

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

Definition

Let $\mathrm{CTop}$ be some small version (…details missing…) of the site of locally contractible contractible topological spaces with continuous maps betwen them and equpped 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 $U\in \mathrm{CTop}$ degreewise by a coproduct of contractibles, also the simplicial set ${\mathrm{lim}}_{\to }Y$ obtained by sending each contractible to a point is contractible. This follows with ArtinMazur. See the proposition below

Define then

$\mathrm{LCTop}\infty \mathrm{Grpd}:={\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\mathrm{CTop}\right)$LCTop\infty Grpd := Sh_{(\infty,1)}(CTop)

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

This is a cohesive (∞,1)-topos.

$\left(\Pi ⊣\mathrm{Disc}⊣\Gamma ⊣\mathrm{coDisc}\right):\mathrm{LCTop}\infty \mathrm{Grpd}\to \infty \mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}}.$(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : LCTop\infty Grpd \to \infty Grpd \,.

Proposition. For $X\in \mathrm{LCTop}↪\mathrm{LC}\infty \mathrm{Grpd}$ a locally contractible space (…maybe with some local size restriction, depending on the details of $\mathrm{CTop}$…), regarded as a 0-truncated locally contractible topological $\infty$-groupoid, we have that the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos-functor applied to $X$ coincides, up to equivalence, with the standard fundamental ∞-groupoid of $X$.

$\Pi \left(X\right)\simeq \mathrm{Sing}X\phantom{\rule{thinmathspace}{0ex}}.$\Pi(X) \simeq Sing X \,.

Proof. By the analogous arguments as at ETop∞Grpd we may present $\Pi$ by the left derived functor of the colimit functor on simplicial presheaves. This is the ordinary colimit applied to a cofibrant resolution of $X$ in $\left[{\mathrm{CTop}}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{proj},\mathrm{loc}}$. By Dugger’s cofibrant replacement theorem recalled at model structure on simplicial presheaves, such is given by a split hypercover $Y\to X$ degreewise a coproduct of objects in $\mathrm{CTop}$. By ArtinMazur there is a weak homotopy equivalence ${\mathrm{lim}}_{\to }Y\to \mathrm{Sing}X$.

(…)

References

• Artin-Mazur, Étale Homotopy , SLNM 100

Revised on September 3, 2012 17:11:10 by Urs Schreiber (131.174.188.82)