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 $BA$:

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 $ℬG$:

See looping and delooping.

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:

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

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

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

• Jacob Lurie, Higher Topos Theory

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

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 $C_1\times C_1\to C_1$ generalizing that of Stasheff H-space from the $(\mathrm{\infty },1)$-category Top to arbitrary $(\mathrm{\infty },1)$-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 ${\mathrm{Hom}}_{BG}(•,•)=G$, or equivalently ${\mathrm{Aut}}_{BG}(•)=G$,

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

More informally but more suggestively we may write

or

to emphasize that there is really only a single object.

Notice how the homotopy pullback works in this simple case:

the universal 2-cell $\eta$

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

to itself, whose component map

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

Related concepts

• loop space object, free loop space object,

  • delooping

  • loop space, free loop space, derived loop space

• suspension object

  • suspension, reduced suspension