nLab
free loop space object

Idea

In an (∞,1)-category $C$ with $(\mathrm{\infty }\mathrm{,1})$-pullbacks, the free loop space object $ℒX$ of any object $X$ is an object that behaves as if its generalized elements are loops in $X$, morphisms between generalized elements homotopies of loops, and so on.

For the case that $C=$ Top this reproduces the ordinary notion of free loop space objects of topological spaces.

Over each fixed element $x\in X$, the free loop space object $ℒX$ looks like the based loop space object ${\Omega }_{x}X$ of $X$.

Free loop space objects come naturally equipped with various structures of interest, such as a categorical circle action. The cohomology of $ℒX$ is Hochschild cohomology or cyclic cohomology of function algebras $C(X)$ on $X$. The categorical circle action induces differentials on these cohomolgies, identifying them, in suitable cases, with algebras of Kähler differential forms on $X$.

Definition

Intrinsic $(\mathrm{\infty }\mathrm{,1})$-categorical definition

Definition

In an (∞,1)-category $C$, for $X\in C$ an object, its free loop space object $ℒX$ is the $(\mathrm{\infty }\mathrm{,1})$-pullback

$$\begin{array}{ccc}ℒX& \to & X\\ \downarrow & & {\downarrow }^{(\mathrm{Id},\mathrm{Id})}\\ X& \stackrel{(\mathrm{Id},\mathrm{Id})}{\to }& X\times X\end{array}.$$\array{ \mathcal{L} X &\to& X \\ \downarrow && \downarrow^{\mathrlap{(Id,Id)}} \\ X &\stackrel{(Id,Id)}{\to}& X \times X } \,.

This is the $(\mathrm{\infty }\mathrm{,1})$-categorical span trace of the identity-span

$$ℒX=\mathrm{Tr}\left(\begin{array}{ccc}& & X\\ & {}^{\mathrm{Id}}\swarrow & & {\searrow }^{\mathrm{Id}}\\ X& & & & X\end{array}\right).$$\mathcal{L}X = Tr \left( \array{ && X \\ & {}^{\mathllap{Id}}\swarrow && \searrow^{\mathrlap{Id}} \\ X &&&& X } \right) \,.

Models by homotopy pullbacks

To see what this amounts to in more detail, assume that the (∞,1)-category is modeled by a homotopical category, say for simplicity a category of fibrant objects, for instance the full subcategory on fibrant objects of a model category.

Then following the discussion at homotopy pullback and generalized universal bundle we can compute the about $(\mathrm{\infty }\mathrm{,1})$-pullback as the ordinary limit

$$\begin{array}{ccccc}ℒX& \to & & \to & X\\ \downarrow & & & & {\downarrow }^{(\mathrm{Id},\mathrm{Id})}\\ & & (X\times X{)}^I& \stackrel{(d_0,d_0)}{\to }& X\times X\\ \downarrow & & {}^{(d_1,d_1)}\downarrow \\ X& \stackrel{(\mathrm{Id},\mathrm{Id})}{\to }& X\times X,\end{array}$$\array{ \mathcal{L}X &\to& &\to& X \\ \downarrow && && \downarrow^{\mathrlap{(Id,Id)}} \\ && (X \times X)^I &\stackrel{(d_0,d_0)}{\to}& X \times X \\ \downarrow && {}^{\mathllap{(d_1,d_1)}}\downarrow \\ X &\stackrel{(Id,Id)}{\to} & X \times X \,, }

where $(X\times X{)}^I$ is a path space object for $X\times X$. At least if we have the structure of a model category we may take $(X\times X{)}^I=X^I\times X^I$ for a path space object $X^I$ of $X$.

From this description one sees that $ℒX$ is built from pairs of paths in $X$ with coinciding endpoints, that are glued at their coinciding endpoint . So the loops here are all built from two semi-ciricle paths.

Properties

Relation to based loop space object

The fiber of $ℒX$ over a point $x:*\to X$ is the corresponding (based) loop space object ${\Omega }_{x}X$ of $X$: we have an $(\mathrm{\infty }\mathrm{,1})$-pullback diagram

$$\begin{array}{ccccc}{\Omega }_{x}X& \to & ℒX& \to & X\\ \downarrow & & \downarrow & & {\downarrow }^{(\mathrm{Id},\mathrm{Id})}\\ *& \stackrel{x}{\to }& X& \stackrel{(\mathrm{Id},\mathrm{Id})}{\to }& X\times X\end{array}.$$\array{ \Omega_x X &\to& \mathcal{L}X &\to& X \\ \downarrow && \downarrow && \downarrow^{\mathrlap{(Id,Id)}} \\ {*} &\stackrel{x}{\to} & X &\stackrel{(Id,Id)}{\to}& X\times X } \,.

To see this, use that homotopy pullbacks paste to homotopy pullbacks, so that the outer pullback is modeled by the ordinary limit

$$\begin{array}{ccccc}{\Omega }_x^{I\vee I}X& \to & & \to & X\\ \downarrow & & & & {\downarrow }^{(\mathrm{Id},\mathrm{Id})}\\ & & (X\times X{)}^I& \stackrel{(d_0,d_0)}{\to }& X\times X\\ \downarrow & & {}^{(d_1,d_1)}\downarrow \\ *& \stackrel{(x,x))}{\to }& X\times X,\end{array}$$\array{ \Omega_x^{I \vee I}X &\to& &\to& X \\ \downarrow && && \downarrow^{\mathrlap{(Id,Id)}} \\ && (X \times X)^I &\stackrel{(d_0,d_0)}{\to}& X \times X \\ \downarrow && {}^{\mathllap{(d_1,d_1)}}\downarrow \\ {*} &\stackrel{(x,x))}{\to} & X \times X \,, }

which builds based loops on $X$ from two consecutive paths, the first starting at the basepoint $x$, the second ending there. This is weakly equivalent ${\Omega }_{x}X={\Omega }_x^{I}X\stackrel{\simeq }{\to }{\Omega }^{I\vee I}X$ to the based loop space object ${\Omega }_{x}X$ built from just the path space object $X^I$ with a single copy of $I$, by standard arguments as for instance form page 12 on in

• Ken Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology .

Categorical circle action

If in addition $C$ has $(\mathrm{\infty }\mathrm{,1})$-pushouts, then a point $x:*\to ℒX$ of the free loop space object is (by definition of limit) the same thing as a map from the pushout

$$S_C^1:=\underrightarrow{\lim }(*\leftarrow *\coprod *\to *)$$S^1_{C} := \lim_\to \left( {*} \leftarrow {*}\coprod{*}\to{*} \right) \,

to $X$. A map $S_C^1\to X$ out of this is, by the nature of homotopy pushouts, two points $x_1,x_2:*\to X$ and two(!) homotopies

$$\left(\begin{array}{cc}& \nearrow {\searrow }^{\simeq }\\ x_1& & x_2\\ & \searrow {\nearrow }_{\simeq }\end{array}\right)\in C$$\left( \array{ & \nearrow \searrow^{\simeq} \\ x_1 && x_2 \\ & \searrow \nearrow_{\simeq} } \right) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \in C

which may be thought of as forming a circle. So $S_C^1$ behaves like a cellular model for the circle with two vertices and two semi-circles. Compare this to the discussion of the nature of $ℒX$ above. This is isomorphic to $*//\mathbb{Z}$, i.e., the classifying space/delooping $B\mathbb{Z}$ in $C$.

Since in Top all these objects are represented by $S^1$, it is reasonable to call them the circle object of the $(\mathrm{\infty }\mathrm{,1})$-category $C$. Using the Yoneda lemma for (∞,1)-categories this means that the free loop space $ℒX$ represents the (∞,1)-presheaf

$${\mathrm{Maps}}_C(S_C^1,X):U\mapsto C(S_C^1\times U,X)$$Maps_C(S^1_{C},X) : U \mapsto C(S^1_C \times U, X)

on $C$. The natural action of the monoid ${\mathrm{End}}_C(S_C^1)$ on $S_C^1$ itself induces an action

$${\mathrm{End}}_C(S_C^1)\times {\mathrm{Maps}}_C(S_C^1,X)\to {\mathrm{Maps}}_C(S_C^1,X)$$End_C(S^1_{C}) \times Maps_C(S^1_{C},X) \to Maps_C(S^1_{C},X)

and, assuming $C$ is closed, again by Yoneda this gives an action

$${\mathrm{End}}_C(S_C^1)\times ℒX\to ℒX.$$End_C(S^1_{C}) \times \mathcal{L}X \to \mathcal{L}X.

Since the action of ${\mathrm{End}}_C(S_C^1)$ on $ℒX$ is just precomposition with a morphism, one sees that if $f:X\to Y$ is a morphism in $C$, then the induced morphism $ℒf:ℒX\to ℒY$ is ${\mathrm{End}}_C(S_C^1)$-equivariant.

Restriction to the subgroup ${\mathrm{Aut}}_{0,C}(S_C^1)\subseteq {\mathrm{End}}_C(S_C^1)$ consisting of automorphisms of $S_C^1$ homotopy equivalent to the identity, gives the categorical circle action

$${\mathrm{Aut}}_{0,C}(S_C^1)\times ℒX\to ℒX$$Aut_{0,C}(S^1_C) \times \mathcal{L}X \to \mathcal{L}X

on $ℒX$. When $C=$ Top this action is the usual rotation of the loops in $X$; indeed, the connected component of the identity in the group of topological automorphisms of $S^1$ is homotopically equivalent to the subgroup of rigid rotations.

Details

Here is what this categorical circle action looks like in more detail:

Urs Schreiber: not clear yet how useful the following is for the above…

we use the delooping groupoid $B\mathbb{Z}$ of the additive group $\mathbb{Z}$ of integers for simplicity, as a model for $S_C^1$. The endomorphisms of this object form the category

$$\mathrm{END}(B\mathbb{Z})$$END(\mathbf{B}\mathbb{Z})

whose objects are functors $B\mathbb{Z}\to B\mathbb{Z}$ and whose morphisms are natural transformations between these. The functors come from group homomorphisms, hence from multiplication by $n\in \mathbb{Z}$

$$[n]:B\mathbb{Z}\to B\mathbb{Z}$$[n] : \mathbf{B}\mathbb{Z} \to \mathbf{B}\mathbb{Z}

$$(•\stackrel{k}{\to }•)\mapsto (•\stackrel{n\cdot k}{\to }•).$$(\bullet \stackrel{k}{\to} \bullet) \mapsto (\bullet \stackrel{n\cdot k}{\to} \bullet).

Natural transformations between two such automorphisms are given by a component $\ell \in \mathbb{Z}$ such that all diagrams

$$\begin{array}{ccc}•& \stackrel{\ell }{\to }& •\\ {}^{n\cdot k}\downarrow & & {\downarrow }^{n\prime \cdot k}\\ •& \stackrel{\ell }{\to }& •\end{array}$$\array{ \bullet &\stackrel{\ell}{\to}& \bullet \\ {}^{\mathllap{n\cdot k}}\downarrow && \downarrow^{\mathrlap{n' \cdot k}} \\ \bullet &\stackrel{\ell}{\to}& \bullet }

commute in $B\mathbb{Z}$. This can happen only for $n=n\prime$, but then it happens for arbitrary $\ell$. In other words we have

$$\mathrm{END}(B\mathbb{Z})\simeq \coprod _{\mathbb{Z}}B\mathbb{Z}.$$END(\mathbf{B}\mathbb{Z}) \simeq \coprod_{\mathbb{Z}}\mathbf{B}\mathbb{Z} \,.

and

$${\mathrm{AUT}}_0(B\mathbb{Z})\simeq B\mathbb{Z}.$$AUT_0(\mathbf{B}\mathbb{Z}) \simeq \mathbf{B}\mathbb{Z} \,.

The object $[n]$ corresponds to the self-mapping of the circle onto itself that fixes the basepoint and has winding number $n\in \mathbb{Z}$. The transformation $\ell$ corresponds then to a rigid rotation of the loop by $\ell$ full circles

Notably for $n=1$ and $k=1$ we may think of the diagram

$$\begin{array}{ccc}•& \stackrel{\ell }{\to }& •\\ {}^1\downarrow & & {\downarrow }^1\\ •& \stackrel{\ell }{\to }& •\end{array}$$\array{ \bullet &\stackrel{\ell}{\to}& \bullet \\ {}^{\mathllap{1}}\downarrow && \downarrow^{\mathrlap{1}} \\ \bullet &\stackrel{\ell}{\to}& \bullet }

as depicting the unit loop around the circle (on the left, say) and the result of translating its basepoint $\ell$-times around the circle (the rest of the diagram). Of course since we are using a model of $S_C^1$ with a single object here, every rotation of the loop is a full circle rotation, which is a bit hard to see.

Exercise: spell out the above discussion analogously for the equivalent model given by the fundamental groupoid ${\Pi }_1(S^1)$ of the standard circle. The is the groupoid with $S_{\mathrm{Top}}^1$ as its set of objects homotopy classes of paths in the circle as morphisms. In this model things look more like one might expect from a circle action. Notice that $B\mathbb{Z}$ is the skeleton of ${\Pi }_1(S^1)$.

David Roberts: I presume that the 2-categories of groupoids and 1-types are equivalent (for the latter I’m taking a subcategory of the usual 2-category of CW complexes, with classes of homotopies as 2-arrows), giving us the conjectural result that the 2-group $\mathrm{AUT}({\Pi }_1(S^1))$ should be equivalent to the 2-group ${\mathrm{AUT}}_{\mathrm{Ho}}(S^1)$ with objects the self-homotopy equivalences of $S^1$ and morphisms the (homotopy classes of) homotopies between them. At the very least, without the above equivalence of 2-categories there should be a faithful functor ${\mathrm{AUT}}_{\mathrm{Ho}}(S^1)\to \mathrm{AUT}({\Pi }_1(S^1))$ which is also injective on objects. A simple exercise should show that it is full.

Urs Schreiber: yes, certainly, these should be equivalent. The entire discussion is maybe a bit of an exercise in “the obvious made explicit to the point that it becomes non-obvious”.

Consider for instance $X=BG$ such that $ℒX=G/{/}_{\mathrm{Ad}}G$ (as discussed in detail below). Then a morphism $(g\stackrel{h}{\to }{\mathrm{Ad}}_{h}a)$ in $G//G$ corresponds to a natural transformation

$$\begin{array}{cc}& \nearrow {\searrow }^g\\ B\mathbb{Z}& {\Downarrow }^h& BG\\ & \searrow {\nearrow }_{{\mathrm{Ad}}_{h}g}\end{array}.$$\array{ & \nearrow \searrow^{\mathrlap{g}} \\ \mathbf{B}\mathbb{Z} &\Downarrow^{h}& \mathbf{B}G \\ & \searrow \nearrow_{\mathrlap{Ad_h g}} } \,.

Precomposing this with the automorphism $\ell$ of the object $[n]$ in $\mathrm{END}(B\mathbb{Z})$

$$\begin{array}{cc}& \nearrow {\searrow }^n\\ B\mathbb{Z}& {\Downarrow }^{\ell }& B\mathbb{Z}\\ & \searrow {\nearrow }_n\end{array}$$\array{ & \nearrow \searrow^{\mathrlap{n}} \\ \mathbf{B}\mathbb{Z} &\Downarrow^{\ell}& \mathbf{B}\mathbb{Z} \\ & \searrow \nearrow_{\mathrlap{n}} }

produces the new transformation

$$\begin{array}{cccc}& \nearrow {\searrow }^n& & \nearrow {\searrow }^g\\ B\mathbb{Z}& {\Downarrow }^{\ell }& B\mathbb{Z}& {\Downarrow }^h& BG\\ & \searrow {\nearrow }_n& & \searrow {\nearrow }_{{\mathrm{Ad}}_{h}g}\end{array}.$$\array{ & \nearrow \searrow^{\mathrlap{n}} && \nearrow \searrow^{\mathrlap{g}} \\ \mathbf{B}\mathbb{Z} &\Downarrow^{\ell}& \mathbf{B}\mathbb{Z} &\Downarrow^{h}& \mathbf{B}G \\ & \searrow \nearrow_{\mathrlap{n}} && \searrow \nearrow_{\mathrlap{Ad_h g}} } \,.

By the rules of horizontal composition of natural transformations, this is the transformation whose component naturality square on $(•\stackrel{1}{\to }•)$ in $B\mathbb{Z}$ is the diagram

$$\begin{array}{ccccc}•& \stackrel{g^{\ell }}{\to }& •& \stackrel{h}{\to }& •\\ {}^{g^n}\downarrow & & {}^{g^n}\downarrow & & {\downarrow }^{{\mathrm{Ad}}_h{g}^n}\\ •& \underset{g^{\ell }}{\to }& •& \underset{h}{\to }& •\end{array}$$\array{ \bullet &\stackrel{g^\ell}{\to}& \bullet &\stackrel{h}{\to}&\bullet \\ {}^{\mathllap{g^{n}}}\downarrow && {}^{g^n}\downarrow && \downarrow^{\mathrlap{Ad_h g^n}} \\ \bullet &\underset{g^\ell}{\to}& \bullet &\underset{h}{\to}&\bullet }

in $B\mathbb{Z}$, hence the morphism $(g^n\stackrel{g^{\ell }h}{\to }{\mathrm{Ad}}_h{g}^n)$ in $G/{/}_{\mathrm{Ad}}G$. In particular, the categorical circle action is

$$\ell :(g\stackrel{h}{\to }{\mathrm{Ad}}_{h}g)\mapsto (g\stackrel{g^{\ell }h}{\to }{\mathrm{Ad}}_{h}g).$$\ell:(g \stackrel{h}{\to} Ad_h g)\mapsto (g \stackrel{g^{\ell} h}{\to} Ad_h g).

Hochschild cohomology and cyclic cohomology

quasicoherent ∞-stacks on $ℒX$ form the Hochschild homology object of $X$ (if the axioms of geometric function theory are met) as described there. The circle acton on $ℒX$ induces differentials on these.

… details to be written, but see Hochschild cohomology and cyclic cohomology for more.

Examples

Free topological loop spaces

• From the above limit description we see that in Top this yields a version for the ordinary free loop space of a topological space.

Details for $ℒBG$

Let the ambient (∞,1)-category be ∞Grpd, let $G$ be an ordinary group and $BG$ its one-object delooping groupoid.

Proposition

We have

$$ℒBG\simeq G/{/}_{\mathrm{Ad}}G,$$\mathcal{L} \mathbf{B}G \simeq G//_{Ad} G \,,

the action groupoid of the adjoint action of $G$ on itself.

Proof

We spell this out in full pedestrian detail, as a little exercise in computing homotopy pullbacks.

We have that the path space object is $B{G}^I=[I,BG]$ – the functor groupoid, where $I$ is the free groupoid $I=\{a\stackrel{\simeq }{\to }b\}$ on the standard interval object – which is (by the definition of natural transformation) the action groupoid

$$B{G}^I=G\backslash \backslash G//G$$\mathbf{B}G^I = G\backslash \backslash G // G

for the action of $G$ on itself, by inverse left and direct right multiplication separately: the naturality square of a natural transformation defining a morphism $g\stackrel{h_1,h_2}{\to }{h}_1^{-1}g{h}_2$ in this groupoid is the commuting square

$$\begin{array}{ccc}•& \stackrel{g}{\to }& •\\ {}^{h_1}\downarrow & & {\downarrow }^{h_2}\\ •& \stackrel{h_1^{-1}g{h}_2}{\to }& •\end{array}$$\array{ \bullet &\stackrel{g}{\to}& \bullet \\ {}^{\mathllap{h_1}}\downarrow && \downarrow^{\mathrlap{h_2}} \\ \bullet &\stackrel{h_1^{-1}g h_2}{\to}& \bullet }

in $BG=*//G$.

The pullback of the top right corner of the above defining limit diagram is

$$\begin{array}{ccc}(G\backslash \backslash G\times G\backslash \backslash G)//G& \to & BG\\ \downarrow & & {\downarrow }^{(\mathrm{Id},\mathrm{Id})}\\ (G\backslash \backslash G//G)\times (G\backslash \backslash G//G)& \to & BG\times BG\end{array}$$\array{ (G\backslash\backslash G \times G\backslash\backslash G)//G &\to& \mathbf{B}G \\ \downarrow && \downarrow^{\mathrlap{(Id,Id)}} \\ (G\backslash\backslash G//G) \times (G\backslash\backslash G//G) &\to& \mathbf{B}G \times \mathbf{B}G }

identifying the two actions from the right, and then the remaining pullback completing the limit diagram is

$$\begin{array}{ccc}G\backslash \backslash (G\times G)//G& \to & (G\backslash \backslash G\times G\backslash \backslash G)//G\\ \downarrow & & \downarrow \\ BG& \stackrel{(\mathrm{Id},\mathrm{Id})}{\to }& BG\times BG\end{array}$$\array{ G\backslash\backslash (G\times G)//G &\to& (G\backslash\backslash G \times G\backslash\backslash G)//G \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{(Id,Id)}{\to}& \mathbf{B}G \times \mathbf{B}G }

now identifying also the two actions from the left, so that $G\backslash \backslash (G\times G)//G$ is the action groupoid of $G$ acting diagonally on $G\times G$ by multiplication from the left and from the right, separately.

To see better what this is, we pass to an equivalent smaller groupoid (the homotopy pullback is defined, of course, only up to weak equivalence). Notice that every morphism $(g_1,g_2)\stackrel{h_1,h_2}{\to }(g{\prime }_1,g{\prime }_2)$ in $G\backslash \backslash (G\times G)//G$ corresponding to a natural transformation

$$\begin{array}{ccc}•& \stackrel{(g_1,g_2)}{\to }& •\\ {}^{(h_1,h_1)}\downarrow & & {\downarrow }^{(h_2,h_2)}\\ •& \stackrel{(h_1^{-1}{g}_1{h}_2,h_1^{-1}{g}_2{h}_2)}{\to }& •\end{array}$$\array{ \bullet &\stackrel{(g_1,g_2)}{\to}& \bullet \\ {}^{\mathllap{(h_1,h_1)}}\downarrow && \downarrow^{\mathrlap{(h_2,h_2)}} \\ \bullet &\stackrel{(h_1^{-1} g_1 h_2, h_1^{-1} g_2 h_2)}{\to} & \bullet }

between functors $I\times I\to BG\times BG$ may always be decomposed as

$$\begin{array}{ccc}•& \stackrel{(g_1,g_2)}{\to }& •\\ {}^{(e,e)}\downarrow & & {\downarrow }^{(g_2^{-1},g_2^{-1})}\\ •& \stackrel{(g_1{g}_2^{-1},e)}{\to }& •\\ {}^{(h_1,h_1)}\downarrow & & {\downarrow }^{(h_1,h_1)}\\ •& \stackrel{(h_1^{-1}(g_1{g}_2^{-1})h_1,e)}{\to }& •\\ {}^{(e,e)}\downarrow & & {\downarrow }^{(g{\prime }_2,g{\prime }_2)}\\ •& \stackrel{(h_1^{-1}(g_1{g}_2^{-1})h_{1}g{\prime }_2,g{\prime }_2)}{\to }& •\end{array}.$$\array{ \bullet &\stackrel{(g_1,g_2)}{\to}& \bullet \\ {}^{\mathllap{(e,e)}}\downarrow && \downarrow^{\mathrlap{(g_2^{-1}, g_2^{-1})}} \\ \bullet &\stackrel{(g_1 g_2^{-1}, e)}{\to}& \bullet \\ {}^{\mathllap{(h_1,h_1)}}\downarrow && \downarrow^{\mathrlap{(h_1,h_1)}} \\ \bullet &\stackrel{(h_1^{-1}(g_1 g_2^{-1})h_1, e)}{\to}& \bullet \\ {}^{\mathllap{(e,e)}}\downarrow && \downarrow^{\mathrlap{(g'_2,g'_2)}} \\ \bullet &\stackrel{(h_1^{-1}(g_1 g_2^{-1})h_1 g'_2, g'_2)}{\to}& \bullet } \,.

Staring at this for a moment shows that this is a unique factorization of every morphism through one of the form

$$\begin{array}{ccc}•& \stackrel{(g,e)}{\to }& •\\ {}^k\downarrow & & {\downarrow }^k\\ •& \stackrel{({\mathrm{Ad}}_{k}g,e)}{\to }& •\end{array},$$\array{ \bullet & \stackrel{(g,e)}{\to} & \bullet \\ {}^{\mathllap{k}}\downarrow && \downarrow^{\mathrlap{k}} \\ \bullet & \stackrel{(Ad_k g,e)}{\to} & \bullet } \,,

which is naturally identified with a morphism in the action groupoid $G/{/}_{\mathrm{Ad}}G$ of the adjoint action of $G$ on itself.

This means that the inclusion

$$G/{/}_{\mathrm{Ad}}G\stackrel{}{\hookrightarrow }G\backslash \backslash (G\times G)//G$$G//_{Ad} G \stackrel{}{\hookrightarrow} G\backslash\backslash(G \times G)//G

given by this identification is essentially surjective and full and faithful, and hence an equivalence of groupoids.

So in conclusion we have that the free loop space object of the delooping $BG$ of a group is

$$ℒBG\simeq G/{/}_{\mathrm{Ad}}G.$$\mathcal{L} \mathbf{B}G \simeq G//_{Ad}G \,.

Chern character

We describe how the Chern character of vector bundles over $X$ may be realized in terms of the cohomology of the free loop space object $ℒX$.

Assume now $C$ is a nice category of smooth spaces, and let $X$ be an object of $C$.

Consider a group object $G$ in $C$ and a representation of $G$ given my a group homomorphism to the general linear group (in $C$): $\rho :G\to \mathrm{GL}(n;ℂ)$. For instance $G$ could be $\mathrm{GL}(n)$ itself and this morphism the identity.

The trace of the representation $\rho$ is invariant under conjugation in the group and so defnes a map $\mathrm{Tr}(\rho ):G/{/}_{\mathrm{Ad}}G\to ℂ$ – a class function. By the equivalence $ℒBG\simeq G/{/}_{\mathrm{Ad}}G$ discussed above, this may be regarded as a characteristic class

$$\mathrm{Tr}(\rho (-)):ℒBG\to ℂ$$Tr(\rho(-)) : \mathcal{L}\mathbf{B}G\to \mathbb {C}

on the free loop space of $BG$.

The cocycle $g:X\to BG$ of a $G$-principal bundle on $X$ transgresses to a cocycle

$$ℒg:ℒX\to ℒBG$$\mathcal{L} g : \mathcal{L}X \to \mathcal{L}\mathbf{B}G

on the free loop space, by the functoriality of the free loop space object construction.

The above characteristic class of this cocycle is the composite morphism

$$\mathrm{Tr}(\rho (ℒg)):ℒX\to ℒBG\to ℂ,$$Tr(\rho(\mathcal{L}g)) : \mathcal{L}X \to \mathcal{L} \mathbf{B}G \to \mathbb{C} \,,

which by the $\mathrm{Ad}$-invariance of the trace is now $S^1$-invariant and hence defines an element in the cyclic cohomology $C(ℒX,ℂ{)}^{S_C^1}$ of $X$.

The Hom-space $C(ℒX,ℂ)$ is a model for the graded commutative algebra of complex-valued differential forms on $X$, with the categorical circle action corresponding to the de Rham differential. Hence $C(ℒX,ℂ{)}^{S_C^1}$ is a model for closed forms and maps to de Rham cohomology $H_{\mathrm{dR}}^{•}(X)$ of $X$. If the de Rham theorem holds for $X$ in $C$, then this may be identified with the real cohomology $H^{•}(X,ℝ)$.

In the case that $G=\mathrm{GL}(\mathrm{\infty };ℂ)$, the compatibility of the trace with direct sums and tensor products of vector bundles over $X$ makes the above construction a ring homomorphism $K(X)\to H_{\mathrm{dR}}(X)$ from the topological K-theory of $X$ to de Rham cohomology, hence a very good candidate to being the Chern character

( to be completed… )

References

Free loop space objects in the (∞,1)-topos of derived stacks on the site of differential graded algebras are discussed in

• David Ben-Zvi, David Nadler, Loop Spaces and Connections (arXiv)