locally contractible topological infinity-groupoid

this page is under construction


Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory



Presentation over a site

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

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?



topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Basic homotopy theory



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


Let CTopCTop 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 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 as pointed out on MO here.1

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 an cohesive (∞,1)-topos.

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


The corresponding 1-cohesive topos over locally connected topological spaces was considered in

  • Peter Johnstone, example 1.5 of Remarks on punctual local connectedness, Theory and Applications of Categories, Vol. 25, 2011, No. 3, pp 51-63 (web)

A decent account of the above \infty-topos is in prepation by David Carchedi

  1. Thanks to David Carchedi for highlighting this.

Revised on January 15, 2016 14:51:22 by Urs Schreiber (