nLab
geometric realization of cohesive infinity-groupoids

Definition

For $(\Pi ⊣\mathrm{Disc}⊣\Gamma ⊣\mathrm{coDisc}):H\to \mathrm{\infty }\mathrm{Grpd}$ a cohesive (∞,1)-topos, we call

$$\mid \Pi (-)\mid :H\stackrel{\Pi }{\to }\mathrm{\infty }\mathrm{Grpd}\stackrel{\mid -\mid }{\to }\mathrm{Top}$${\vert \Pi (- )\vert} : \mathbf{H} \stackrel{\Pi}{\to} \infty Grpd \stackrel{\vert - \vert}{\to} Top

the geometric realization functor. For $X\in H$ any object, hence any cohesive ∞-groupoid, $\mid \Pi (X)\mid$ is its geometric realization.

Notice that $\Pi (X)$ is the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos and $\mid -\mid :$ ∞Grpd $\to$ Top is the “homotopy hypothesis” equivalence of (∞,1)-categories.

Properties

See at cohesive (∞,1)-topos -- structures the section Geometric homotopy and Galois theory.

Examples

In $H=$ ETop∞Grpd the geometric realization of cohesive $\mathrm{\infty }$-groupoids subsumes the geometric realization of simplicial topological spaces (see there for details).

Related concepts

• geometric realization

  • of categories, of simplicial topological spaces, of cohesive ∞-groupoids