#
nLab
geometric realization of cohesive infinity-groupoids

### Context

#### Cohesive $\infty$-Toposes

#### Homotopy theory

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

# Contents

## 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).