Stabe homotopy theory
The delooping of an object is, if it exists, a uniquely pointed object such that is the loop space object of :
In particular, if is a group then its delooping
Under the homotopy hypothesis these two objects are identified: the geometric realization of the groupoid is the classifying space :
See looping and delooping.
Loop space objects are defined in any (∞,1)-category with homotopy pullbacks: for any pointed object of with point , its loop space object is the homotopy pullback of this point along itself:
Conversely, if is given and a homotopy pullback diagram
exists, with the point being essentially unique, by the above has been realized as the loop space object of
and we say that is the delooping of .
See the section delooping at groupoid object in an (∞,1)-category for more.
If is even a stable (∞,1)-category then all deloopings exist and are then also denoted and called the suspension of .
Characterization of deloopable objects
In section 6.1.3 of
a definition of groupoid object in an (infinity,1)-category is given as a homotopy simplicial object, i.e. a (infinity,1)-functor
satisfying certain conditions (prop. 22.214.171.124) which are such that if is the point we have an internal group in a homotopical sense, given by an object equipped with a coherently associative multiplication operation generalizing that of Stasheff H-space from the -category Top to arbitrary -categories.
Lurie calls the groupoid object an effective groupoid object in an (infinity,1)-category precisely if it arises as the delooping, in the above sense, of some object .
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.
Topological loop spaces
For 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 be a group regarded as a discrete groupoid in the (∞,1)-topos ∞Grpd of ∞-groupoids.
Then exists and is, up to equivalence, the groupoid
with a single object ,
with , or equivalently ,
and with composition of morphisms in being given by the product operation in the group.
More informally but more suggestively we may write
to emphasize that there is really only a single object.
Notice how the homotopy pullback works in this simple case:
the universal 2-cell
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 and .