nLab
locally contractible topological infinity-groupoid

this page is under construction

Context

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

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

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

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 LCTopGrpdLCTop\infty Grpd.

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

Definition

Let CTopCTop 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 YUY \to U a hypercover of UCTopU \in CTop degreewise by a coproduct of contractibles, also the simplicial set lim Y\lim_\to Y obtained by sending each contractible to a point is contractible. This follows with ArtinMazur. See the proposition below

Define then

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

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

This is a cohesive (∞,1)-topos.

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

Proposition. For XLCTopLCGrpdX \in LCTop \hookrightarrow LC\infty Grpd a locally contractible space (…maybe with some local size restriction, depending on the details of CTopCTop…), 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 XX coincides, up to equivalence, with the standard fundamental ∞-groupoid of XX.

Π(X)SingX. \Pi(X) \simeq Sing X \,.

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

(…)

References

  • Artin-Mazur, Étale Homotopy , SLNM 100

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