nLab geometric realization of cohesive infinity-groupoids

Context

Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Structures in a cohesive $(\infty,1)$-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

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 hypothesisequivalence 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).

Revised on October 11, 2013 23:14:44 by Urs Schreiber (89.204.135.160)