nLab
geometric realization of cohesive infinity-groupoids

Context

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

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

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Homotopy theory

Contents

Definition

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

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

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

Notice that Π(X)\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 H=\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)