nLab topological infinity-groupoid

Idea

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 $Top_{sm}$ 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:

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

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 $Top_{sm}$ to make. For instance

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

2. taking $Top_{sm}$ 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.

Related concepts

• discrete ∞-groupoid

• D-topological ∞-groupoid

• smooth ∞-groupoid

• formal smooth ∞-groupoid

• super formal smooth ∞-groupoid