nLab
topological infinity-groupoid

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

A topological ∞-groupoid is meant to be an ∞-groupoid that is equipped with topological structure. For instance a 0-truncated topological \infty-groupoid should just be a topological space, and 1-truncated topological \infty-groupoids should reproduce topological groupoids/topological stacks, etc.

A generic way to make sense of this in a gros topos perspective is to pick some small subcategory Top smTop_{sm} of the category Top of topological spaces, regard it as a site with respect to an evident coverage by open covers, and then take the (∞,1)-topos of generalized topological \infty-groupoids to be the (∞,1)-category of (∞,1)-sheaves on this site:

TopGrpdSh (Top sm). Top \infty Grpd \coloneqq Sh_\infty(Top_{sm}) \,.

This gives a nice category with very general objects. In there one may find smaller, less nice categories of nicer objects.

There are different choices of sites Top smTop_{sm} to make. For instance

  1. taking Top smTop_{sm} to be a small site of locally contractible topological spaces yields a concept of locally contractible topological infinity-groupoids;

  2. taking Top smTop_{sm} be the the subcategory of topological Cartesian spaces yield a concept of Euclidean-topological infinity-groupoids.

These two hapenns to constitute cohesive ∞-toposes, due to the local contractibility of the objects in the site.

Revised on September 11, 2017 17:32:13 by David Corfield (209.93.157.137)