nLab loop space object

Idea

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.

Definition

Loop space objects are defined in any (∞,1)-category $\mathbf{C}$ with homotopy pullbacks: for $X$ any pointed object of $\mathbf{C}$ with point ${*} \to X$, its loop space object is the homotopy pullback $\Omega X$ of this point along itself:

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

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

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

Remarks

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

See also

• fibration sequence

• delooping

• stable (∞,1)-category

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 $* \xrightarrow{0} I \xleftarrow{1} *$ the free loop space object is the part of the path object $B^I = [I,B]$ which consists of closed paths, namely the pullback

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

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

$$\left[ \array{ && cotr(I) \\ & \nearrow && \nwarrow \\ pt &&&& I \\ & {}_{Id \sqcup Id}\nwarrow && \nearrow_{in \sqcup out} \\ && pt \sqcup pt } \;\;\;\;,\;\;\;\; B \right] \;,\;\;\;\; \simeq \;,\;\;\;\; \array{ && \Lambda B \\ & \swarrow && \searrow \\ B &&&& [I,B] \\ & {}_{Id \times Id}\searrow && \swarrow_{d_0 \times d_1} \\ && B \times B }$$

Based loop space objects

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

$$\array{ \Omega_{pt}B &\to& [I,B] \\ \downarrow && \downarrow^{d_0 \times d_1} \\ pt &\stackrel{pt_B \times pt_B}{\to}& B \times B } \,.$$

Remarks

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

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

  $$\array{ \Omega_{pt} B &\to& \Lambda B \\ \downarrow && \downarrow \\ pt &\stackrel{pt_B}{\to}& B } \,.$$

Remarks

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

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

• In a suitable extension of $\operatorname{Diff}$, 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.

Examples

• Let $C =$ Top with the standard interval object. Then for $B= X$ a topological space $\Lambda B = \Lambda X$ is the ordinary free loop space of $X$.

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

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

  $$\Lambda \mathbf{B}G = G//_{Ad}G$$

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

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

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

  • Bruce Bartlett, On unitary 2-representations of finite groups and topological quantum field theory (arXiv)

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

  $$\Omega \mathbf{B}G = G \,.$$

Related concepts

• loop space object, free loop space object

  loop space type

  • delooping, looping and delooping

  • loop space, free loop space, derived loop space

  • inertia orbifold

• suspension object

  • suspension, reduced suspension