loop space object



Stabe homotopy theory

Mapping space

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 C\mathbf{C} with homotopy pullbacks: for XX any pointed object of C\mathbf{C} with point *X{*} \to X, its loop space object is the homotopy pullback ΩX\Omega X of this point along itself:

ΩX * * X. \array{ \Omega X &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& X } \,.

A (generalised) element of ΩX\Omega X may be thought of as a loop in XX at the base point **.


Since C(X,)\mathbf{C}(X,-) commutes with homotopy limits, one has a natural homotopy equivalence ΩC(X,Y)C(X,ΩY)\Omega\mathbf{C}(X,Y)\simeq \mathbf{C}(X,\Omega Y), for any two objects XX and YY in C\mathbf{C}.

See also

Explicit constructions

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 B I=[I,B]B^I = [I,B] which consists of closed paths, namely the pullback

ΛB [I,B] d 0×d 1 B Id×Id B×B. \array{ \Lambda B &\to& [I,B] \\ \downarrow && \downarrow^{\mathrlap{d_0 \times d_1}} \\ B &\stackrel{Id \times Id}{\to}& B \times B } \,.

This is the same as the image of the co-span co-trace cotr(I)cotr(I) of the interval object (which is the interval object closed to a loop!, see the examples at co-span co-trace) in BB:

[ cotr(I) pt I IdId inout ptpt,B],, ΛB B [I,B] Id×Id d 0×d 1 B×B \left[ \array{ && cotr(I) \\ & \nearrow && \nwarrow \\ pt &&&& I \\ & {}_{Id \sqcup Id}\nwarrow && \nearrow_{in \sqcup out} \\ && pt \sqcup pt } \;\;\;\;,\;\;\;\; B \right] \;,\;\;\;\; \simeq \;,\;\;\;\; \array{ && \Lambda B \\ & \swarrow && \searrow \\ B &&&& [I,B] \\ & {}_{Id \times Id}\searrow && \swarrow_{d_0 \times d_1} \\ && B \times B }

Based loop space objects

If BB is a pointed object with point ptpt BBpt \stackrel{pt_B}{\to} B then the based loop space object of BB is the pullback Ω ptB\Omega_{pt} B in

Ω ptB [I,B] d 0×d 1 pt pt B×pt B B×B. \array{ \Omega_{pt}B &\to& [I,B] \\ \downarrow && \downarrow^{d_0 \times d_1} \\ pt &\stackrel{pt_B \times pt_B}{\to}& B \times B } \,.


  • Ω ptB\Omega_{pt}B is the fiber of the generalized universal bundle E ptBB\mathbf{E}_{pt}B \to B.

  • the based loop space object Ω ptB\Omega_{pt} B is the pullback of the free loop space object ΛB\Lambda B to the point

    Ω ptB ΛB pt pt B B. \array{ \Omega_{pt} B &\to& \Lambda B \\ \downarrow && \downarrow \\ pt &\stackrel{pt_B}{\to}& B } \,.


  • The loop space object BB can be regarded as the homotopy trace on the identity span on BB, as described at span trace.

  • The free loop space object inherits the structure of an A A_\infty-category from that of the path object [I,B][I,B].

  • In a suitable extension of Diff\operatorname{Diff}, 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 C=C = Top with the standard interval object. Then for B=XB= X a topological space ΛB=ΛX\Lambda B = \Lambda X is the ordinary free loop space of XX.

    The generalization of this to smooth spaces is discussed at smooth loop space.

  • Let C=C = Grpd with the standard interval object I={ab}I = \{a \stackrel{\simeq}{\to} b\} and let BG\mathbf{B}G be the one-object groupoid corresponding to a group GG, then

    ΛBG=G// AdG \Lambda \mathbf{B}G = G//_{Ad}G

    is the action groupoid of GG acting on itself by its adjoint action. Notice the example at co-span co-trace which says that the cotrace on II is cotr(I)=Bcotr(I) = \mathbf{B}\mathbb{Z}, and indeed

    ΛBG=[B,BG]. \Lambda \mathbf{B}G = [\mathbf{B}\mathbb{Z}, \mathbf{B}G] \,.

    The role of this ΛBG\Lambda \mathbf{B}G 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)


  • On the other hand, the based loop object of BG\mathbf{B}G is just GG:

    ΩBG=G. \Omega \mathbf{B}G = G \,.

Revised on October 20, 2015 05:23:58 by David Corfield (