Contents

Idea

In an (∞,1)-category $C$ with $(\infty,1)$-pullbacks, the free loop space object $\mathcal{L}X$ of any object $X$ – also called the inertia groupoid – 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 spaces of topological spaces.

Over each fixed element $x \in X$, the free loop space object $\mathcal{L}X$ 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 $\mathcal{L}X$ is Hochschild cohomology or cyclic cohomology of function algebras $C(X)$ on $X$. The categorical circle action induces differentials on these cohomologies, identifying them, in suitable cases, with algebras of Kähler differential forms on $X$.

Definition

In higher category theory

There are various equivalent ways to define the free loop space object.

Let $C$ be an (∞,1)-category. Recall that every $(\infty,1)$-category is enriched over ?Gpd?, in that there is a hom-space $Map(X,Y) \in \infty Gpd$ for any $X,Y\in C$. This enables us to define the power of an object $X\in C$ by any $\infty$-groupoid $K$, as an object $X^K \in C$ together with a map $K \to Map(X^K,X)$ inducing equivalences $Map(Y,X)^K \simeq Map(Y,X^K)$ for all $Y\in C$ (where $Map(Y,X)^K$ denotes the mapping-space from $K$ to $Map(Y,X)$ in $\infty Gpd$), if such exists.

This can also be written in terms of “conical” limits in $C$. Most commonly, if we note that $S^1$ is the pushout of two copies of $\ast$ under $\ast\sqcup \ast$, and that $Map(\ast,X) \simeq X$ while $Map(\ast\sqcup \ast ,X) \simeq X\times X$, we find that $X^{S^1}$ is the pullback of $X$ and $X$ over $X\times X$:

We can also use the fact that $S^1$ is the homotopy coequalizer of $\ast \rightrightarrows \ast$:

In homotopy type theory

In the literature (see below) when the free loop space object $\mathcal{L}X$ is defined as the homotopy pullback $X\times_{X\times X} X$, it is sometimes described heuristically as: “a point of $\mathcal{L}X$ is a choice of making two points of $X$ equal in two ways.” In terms of homotopy type theory this heuristics becomes a theorem. In that higher categorical logic we have the expression

Here on the right we have

1. the dependent sum;

2. over the identity type $Id (X \times X)$;

3. of the product type $X \times X$.

See the discussion at homotopy pullback in the section Construction in homotopy type theory for how this is equivalent to the previous definition.

Now since $\sum_{y:X} (x=y)$ is contractible, the above type is equivalent to

i.e. the type of points that are equal to themselves (in a specified, not necessarily reflexivity, way). This corresponds to the other definitions, as a homotopy equalizer or a powering by $S^1$.

Properties

Models by homotopy pullbacks

To see what the definition of a free loop space object 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 $(\infty,1)$-pullback $X\times_{X\times X}^h X$ as the ordinary limit

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 $\mathcal{L}X$ 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.

Relation to based loop space object

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

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

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 $\Omega_x X = \Omega^I_x X \stackrel{\simeq}{\to} \Omega^{I \vee I} X$ 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

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

Action on objects of the slice

The free loop space object $\mathcal{L} X$ is a group object in the slice (∞,1)-category $C/X$, and has a canonical action on all objects of $C/X$.

Intuitively, the group structure comes from composition and inversion of loops. When the free loop space is expressed as a power $X^{S^1}$, this group structure comes from the canonical cogroup structure on $S^1$. In homotopy type theory, it is literally concatenation of paths. And when $\mathcal{L}X$ is expressed as the pullback $X\times_{X\times X} X$, the group multiplication can be obtained by considering the following pasting square:

Here the bottom-left and top-right squares are the pullback defining $\mathcal{L}X$, the top-left square is the pullback defining $\mathcal{L}X \times_X\mathcal{L}X$, and the bottom-right square commutes but is not a pullback. By the universal property of the pullback square defining $\mathcal{L}X$, this square factors through it uniquely, giving the composition map $\mathcal{L}X \times_X \mathcal{L}X\to \mathcal{L}X$. Similarly but more simply, the inversion map $\mathcal{L}X\to \mathcal{L}X$ comes from transposing its defining pullback square and factoring it through itself.

Now consider $Y\to X$ an object of $C/X$. There is a canonical projection $\mathcal{L}X \times_X Y \to Y$, which is not the action of $\mathcal{L}X$ on $Y$, but it almost is. In fact since this projection is a morphism in the “homotopy” slice $C/X$, it consists not just of a morphism in $C$ but a homotopy witnessing that a certain triangle commutes, which is equivalently the homotopy in the left-hand commutative square below (which is the pullback defining $\mathcal{L}X \times_X Y$):

Pasting this with the right-hand commutative square, which is the canonical automorphism of the projection $\mathcal{L}X\to X$, we obtain a different homotopy witnessing a different morphism $\mathcal{L}X \times_X Y \to Y$ in $C/X$ (with the same underlying morphism in $C$), and this is the action of $\mathcal{L}X$ on $Y$.

The definitiong of the right-hand commutative square above may not be obvious. It is clear when we write $\mathcal{L}X = X^{S^1}$; when we write $\mathcal{L}X = X\times_{X\times X} X$ it can be obtained as the following pasting square, where $p$ and $q$ are the two projections in the defining pullback of $\mathcal{L}X$:

To extend these structures to a coherent $\infty$-group structure and a coherent $\infty$-action, see for instance this MO answer.

In an $(\infty,1)$-topos

We consider now the case that $C = \mathbf{H}$ is an (∞,1)-topos (of (∞,1)-sheaves/∞-stacks). This comes canonically with its terminal global sections (∞,1)-geometric morphism

In this case we can reformulate the power definition $\mathcal{L}X = X^{S^1}$ using a version of $S^1$ that is an object of $\mathbf{H}$ itself.

As a mapping space object

Intrinsic circle action

By precomposition, the automorphism 2-group of the circle $S^1$ acts on free loop space of an object $X \in \mathbf{H}$

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 $S^1_{Top}$ 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 $\mathbf{B}\mathbb{Z}$ is the skeleton of $\Pi_1(S^1)$.

Hochschild cohomology and cyclic cohomology

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

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

Examples

Free topological loop spaces

In Top the notion of free loop space objects reproduces the standard notion of topological free loop spaces.

Details for $\mathcal{L} \mathbf{B}G$

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

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 $\mathcal{L}X$.

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 by a group homomorphism to the general linear group (in $C$): $\rho:G\to GL(n;\mathbb{C})$. For instance $G$ could be $GL(n)$ itself and this morphism the identity.

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

on the free loop space of $\mathbf{B}G$.

The cocycle $g : X\to \mathbf{B}G$ of a $G$-principal bundle on $X$ transgresses to a cocycle

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

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

The Hom-space $C(\mathcal{L}X,\mathbb{C})$ 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 $C(\mathcal{L}X,\mathbb{C})^{S^1_C}$ is a model for closed forms and maps to de Rham cohomology $H_{dR}^\bullet(X)$ of $X$. If the de Rham theorem holds for $X$ in $C$, then this may be identified with the real cohomology $H^\bullet(X,\mathbb{R})$.

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

( to be completed… )

Isotropy of a topos

The isotropy group of a topos is its free loop space object in the 2-category Topos.

Related concepts

• loop space object, free loop space object,

  • delooping

  • loop space, free loop space, derived loop space

• suspension object

  • suspension

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)

More information in the topological case is given in:

• Ronnie BrownCrossed modules and the homotopy 2-type of a free loop space, (arXiv)

which gives complete information on the 2-type of $\mathcal{L}X$ for a space $X$ which is the classifying space of a crossed module of groups. This generalises the above example of $\mathcal{L} \mathbf{B}G$.