nLab geometric realization of cohesive infinity-groupoids

Definition

For $(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \mathbf{H} \to \infty Grpd$ a cohesive (∞,1)-topos, we call the action of the shape modality

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

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

Notice that $\Pi(X)$ is the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos and $\vert - \vert :$ ∞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 $\mathbf{H} =$ ETop∞Grpd the geometric realization of cohesive $\infty$-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

  • shape, shape modality