loop space object


(,1)(\infty,1)-Category theory

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 **.

When the point x:*Xx : {*} \to X is not clear from context, we can write Ω xX\Omega_x X or Ω(X,x)\Omega(X,x) to indicate the point.


Since C(X,)\mathbf{C}(X,-) commutes with homotopy limits, one has a natural homotopy equivalence Ω y¯C(X,Y)C(X,Ω yY)\Omega_{\bar{y}}\mathbf{C}(X,Y)\simeq \mathbf{C}(X,\Omega_y Y), for any objects XX and pointed object (Y,y)(Y,y) in C\mathbf{C}, where y¯\bar{y} denotes the morphism X*YX \to * \to Y.

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 *0T1* * \xrightarrow{0} T \xleftarrow{1} * 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

where d 0 d_0 (d 1d_1 resp.) is the composition of [0,B] [0,B] ([1,B][1,B] resp.) with the canonical identification of [*,B][*, B] and BB.

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 } \,.