Stabe homotopy theory
Stable homotopy theory
Loop space objects
In the (∞,1)-topos Top the construction of a loop space of a given topological space is familiar.
This construction may be generalized to any other (∞,1)-topos and in fact to any other (∞,1)-category with homotopy pullbacks.
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:
A (generalised) element of may be thought of as a loop in at the base point .
Since commutes with homotopy limits, one has a natural homotopy equivalence , for any two objects and in .
Usually the (∞,1)-category in question is presented by concrete 1-categorical data, such as that of a model category. In that case the above homotopy pullback has various realizations as an ordinary pullback.
Notably it may be expressed using path objects which may come from interval objects. Even if the context is not (or not manifestly) that of a homotopical category, an interval object may still exist and may be used as indicated in the following to construct loop space objects.
Free loop space objects
In a category with interval object the free loop space object is the part of the path object which consists of closed paths, namely the pullback
This is the same as the image of the co-span co-trace of the interval object (which is the interval object closed to a loop!, see the examples at co-span co-trace) in :
Based loop space objects
If is a pointed object with point then the based loop space object of is the pullback in
is the fiber of the generalized universal bundle .
the based loop space object is the pullback of the free loop space object to the point
The loop space object can be regarded as the homotopy trace on the identity span on , as described at span trace.
The free loop space object inherits the structure of an -category from that of the path object .
In a suitable extension of , this construction does not give the usual smooth loop space (free or based). It gives the space of paths with coincident endpoints rather than the space of smooth maps from the circle. Thus the smooth loop space is not a loop space object.
Let Top with the standard interval object. Then for a topological space is the ordinary free loop space of .
The generalization of this to smooth spaces is discussed at smooth loop space.
Let Grpd with the standard interval object and let be the one-object groupoid corresponding to a group , then
is the action groupoid of acting on itself by its adjoint action. Notice the example at co-span co-trace which says that the cotrace on is , and indeed
The role of this as a loop object is amplified in particular in
Simon Willerton, The twisted Drinfeld double of a finite group via gerbes and finite groupoids (arXiv)
Bruce Bartlett, On unitary 2-representations of finite groups and topological quantum field theory (arXiv)
On the other hand, the based loop object of is just :