nLab topological infinity-groupoid




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos



A topological ∞-groupoid is meant to be an ∞-groupoid that is equipped with topological structure. For instance a 0-truncated topological \infty-groupoid should just be a topological space, and 1-truncated topological \infty-groupoids should reproduce topological groupoids/topological stacks, etc.

A generic way to make sense of this in a gros topos perspective is to pick some small subcategory Top smTop_{sm} of the category Top of topological spaces, regard it as a site with respect to an evident coverage by open covers, and then take the (∞,1)-topos of generalized topological \infty-groupoids to be the (∞,1)-category of (∞,1)-sheaves on this site:

TopGrpdSh (Top sm). Top \infty Grpd \coloneqq Sh_\infty(Top_{sm}) \,.

This gives a nice category with very general objects. In there one may find smaller, less nice categories of nicer objects.

There are different choices of sites Top smTop_{sm} to make. For instance

  1. taking Top smTop_{sm} to be a small site of locally contractible topological spaces yields a concept of locally contractible topological infinity-groupoids;

  2. taking Top smTop_{sm} be the the subcategory of topological Cartesian spaces yield a concept of Euclidean-topological infinity-groupoids.

These two happen to constitute cohesive ∞-toposes, due to the local contractibility of the objects in the site.

Last revised on September 23, 2021 at 18:02:13. See the history of this page for a list of all contributions to it.