# nLab delooping

### Context

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

## Models

#### Stabe homotopy theory

stable homotopy theory

# Contents

## Idea

The delooping of an object $A$ is, if it exists, a uniquely pointed object $BA$ such that $A$ is the loop space object of $BA$:

$A\simeq \Omega \left(BA\right)$A \simeq \Omega(\mathbf{B} A)

In particular, if $A=G$ is a group then its delooping

• in the context Top is the classifying space $ℬG$

• in the context ∞-Grpd is the one-object groupoid $BG$.

Under the homotopy hypothesis these two objects are identified: the geometric realization of the groupoid $BG$ is the classifying space $ℬG$:

$\mid BG\mid \simeq ℬG\phantom{\rule{thinmathspace}{0ex}}.$|\mathbf{B}G| \simeq \mathcal{B}G \,.

## Definition

Loop space objects are defined in any (∞,1)-category $C$ with homotopy pullbacks: for $X$ any pointed object of $C$ with point $*\to X$, its loop space object is the homotopy pullback $\Omega X$ of this point along itself:

$\begin{array}{ccc}\Omega X& \to & *\\ ↓& & ↓\\ *& \to & X\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \Omega X &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& X } \,.

Conversely, if $A$ is given and a homotopy pullback diagram

$\begin{array}{ccc}A& \to & *\\ ↓& & ↓\\ *& \to & BA\end{array}$\array{ A &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& \mathbf{B}A }

exists, with the point $*\to BA$ being essentially unique, by the above $A$ has been realized as the loop space object of $BA$

$A=\Omega BA$A = \Omega \mathbf{B} A

and we say that $BA$ is the delooping of $A$.

See the section delooping at groupoid object in an (∞,1)-category for more.

## Remarks

If $C$ is even a stable (∞,1)-category then all deloopings exist and are then also denoted $\Sigma A$ and called the suspension of $A$.

## Characterization of deloopable objects

In section 6.1.3 of

a definition of groupoid object in an (infinity,1)-category $C$ is given as a homotopy simplicial objects, i.e. a (infinity,1)-functor

$C:{\Delta }^{\mathrm{op}}\to C$C : \Delta^{op} \to \mathbf{C}
$\cdots {C}_{2}\stackrel{\to }{⇉}{C}_{1}⇉{C}_{0}$\cdots C_2 \stackrel{\to}\rightrightarrows C_1 \rightrightarrows C_0

satisfying certain conditions (prop. 6.1.2.6) which are such that if ${C}_{0}=*$ is the point we have an internal group in a homotopical sense, given by an object ${C}_{1}$ equipped with a coherently associative multiplication operation ${C}_{1}×{C}_{1}\to {C}_{1}$ generalizing that of Stasheff H-space from the $\left(\infty ,1\right)$-category Top to arbitrary $\left(\infty ,1\right)$-categories.

Lurie calls the groupoid object $C$ an effective groupoid object in an (infinity,1)-category precisely if it arises as the delooping, in the above sense, of some object $BC$.

One of the characterizing properties of an (infinity,1)-topos is that every groupoid object in it is effective.

This is the analog of Stasheff’s classical result about H-spaces.

See the remark at the very end of section 6.1.2 in HTT.

## Examples

### Topological loop spaces

For $C=$ Top the (infinity,1)-category of topological spaces, a space is deloopable if it is an A-infinity-space and hence homotopy equivalent to a loop space.

### Delooping of a group to a groupoid

Let $G$ be a group regarded as a discrete groupoid in the (∞,1)-topos ∞Grpd of ∞-groupoids.

Then $BG$ exists and is, up to equivalence, the groupoid

• with a single object $•$,

• with ${\mathrm{Hom}}_{BG}\left(•,•\right)=G$, or equivalently ${\mathrm{Aut}}_{BG}\left(•\right)=G$,

• and with composition of morphisms in $BG$ being given by the product operation in the group.

More informally but more suggestively we may write

$BG=\left\{•\stackrel{g}{\to }•\mid g\in G\right\}$\mathbf{B} G = \{ \bullet \stackrel{g}{\to} \bullet | g \in G\}

or

$BG=\left\{•⟲g\phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}g\in G\right\}$\mathbf{B}G = \{ \bullet \righttoleftarrow g \;|\; g \in G \}

to emphasize that there is really only a single object.

Notice how the homotopy pullback works in this simple case:

the universal 2-cell $\eta$

$\begin{array}{ccc}G& \to & *\\ ↓& {⇓}^{\eta }& ↓\\ *& \to & BG\end{array}$\array{ G &\to& {*} \\ \downarrow &\Downarrow^{\eta}& \downarrow \\ {*} &\to& \mathbf{B}G }

filling this 2-limit diagram is the natural transformation from the constant functor

$G\to *\to BG$G \to {*} \to \mathbf{B}G

to itself, whose component map

$\eta :\mathrm{Obj}\left(G\right)\to \mathrm{Mor}\left(BG\right)$\eta : Obj(G) \to Mor(\mathbf{B}G)

is just the identity map, using that $\mathrm{Obj}\left(G\right)=G$ and $\mathrm{Mor}\left(BG\right)=G$.

Revised on February 3, 2013 15:43:11 by Tim Porter (95.147.236.249)