equivalences in/of $(\infty,1)$-categories
hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
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 $X$.
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.
The free loop space object of $X\in C$ is the power $\mathcal{L}X = X^{S^1}$, if it exists, where $S^1$ denotes the homotopical circle.
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 $X\times X$:
In an (∞,1)-category $C$ with (∞,1)-pullbacks, for $X \in C$ an object, its free loop space object $\mathcal{L}X$ is the $(\infty,1)$-pullback of the diagonal along itself
This is the $(\infty,1)$-categorical span trace of the identity-span
We can also use the fact that $S^1$ is the homotopy coequalizer? of $\ast \rightrightarrows \ast$:
The free loop space object of $X\in C$ is the homotopy equalizer of two copies of the identity map $X \rightrightarrows X$.
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
the dependent sum;
over the identity type $Id (X \times X)$;
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 $S^1$.
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.
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
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.
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.
Write
for the circle. In Top this is the usual topological circle. In ∞Grpd this is (the homotopy type of) the fundamental ∞-groupoid of the topological circle. We may think of this as the (∞,1)-pushout
hence as the universal cocone
in $\infty Grpd$.
In $\mathbf{H}$ we still write $S^1$ for the constant ∞-stack on this, the image of this under $LConst$. Since $LConst$ is a left adjoint and hence preserves this poushout, there is no risk of confusion.
To see that the given $(\infty,1)$-pushout indeed produces the circle, we use the standard model structure on simplicial sets $sSet_{Quillen}$ to present ∞Grpd. In $sSet_{Quillen}$ the $(\infty,1)$-pushout is computed by the homotopy pushout. By general facts about this, it may be computed as an ordinary pushout in sSet once we pass to an equivalent pushout diagram in which at least one morphism is a monomorphism. This is the case for
Informally the $(\infty,1)$-pushout $* \coprod_{* \coprod *} *$ may be thought of as
the disjoint union of two points $*_1$, $*_2$;
equipped with two non-equivalent abstract homotopies between them
This equivalent way of modelling the circle not as a single point with an automorphism, but as two points with two isomorphisms is what connects directly to the definition of the free loop space object. This we now come to. It is also the fundamenal source of the basic structure of Hochschild (co)homology (as discussed there).
Every (∞,1)-topos is a cartesian closed (∞,1)-category: we have for every object $X \in \mathbf{H}$ an internal hom-(∞,1)-functor
This is discussed at (∞,1)-topos in the section Closed monoidal structure.
This follows from the above by the fact (see closed monoidal structure on (∞,1)-toposes) that the internal hom in an $(\infty,1)$-topos preserves finite colimits in its first argument and satisfies
This yields
We have that the free loop space object of $X \in \mathbf{H}$ is equivalently the powering of $X$ by the $\infty$-groupoid $S^1$:
Follows by the above from the equivalence $[LConst S^1 , X] \simeq X^{S^1}$ discussed at (∞,1)-topos.
By precomposition, the automorphism 2-group of the circle $S^1$ acts on free loop space of an object $X \in \mathbf{H}$
The connected component of $[S^1,S^1]$ on the identity is equivalent to $S^1$
We say that
is the intrinsic circle action on the free loop space object.
We spell out in detail what this action looks like. The reader should thoughout keep the homotopy hypothesis-equivalence, $(|-| \dashv \Pi) : Top \simeq \infty Grpd$ in mind.
We may realize the circle $S^1 \in$ Top under $\Pi : Top \simeq \infty Grpd$ as the [delooping]] groupoid $\mathbf{B}\mathbb{Z}$ of the additive group $\mathbb{Z}$ of integers
The automorphism 2-group of this object is the functor groupoid
whose objects are invertible functors $\mathbf{B}\mathbb{Z} \to \mathbf{B}\mathbb{Z}$ and whose morphisms are natural transformations between these.
The functors $\mathbf{B}\mathbb{Z} \to \mathbf{B}\mathbb{Z}$ correspond bijectively to group homomorphisms $\mathbb{Z} \to \mathbb{Z}$, hence to multiplication by $n\in\mathbb{Z}$
Natural transformations between two such endomorphisms are given by a component $\ell \in \mathbb{Z}$ such that all diagrams
commute in $\mathbf{B}\mathbb{Z}$. This can happen only for $n = n'$, but then it happens for arbitrary $\ell$.
In other words we have
and
The object $[n]$ corresponds to the self-mapping of the circle 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
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 $S^1$ 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 $S^1_{Top}$ 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 $\Pi_1(S^1)$.
Consider $\mathbf{H} =$ ∞Grpd, $G$ a group and $X = \mathbf{B}G$ the delooping groupoid?. Then $\mathcal{L}X = G//_{Ad}G$ (as discussed in detail below). A morphism $(g \stackrel{h}{\to} Ad_h a)$ in $G//G$ corresponds to a natural transformation
Precomposing this with the automorphism $\ell$ of the object $[n]$ in $END(\mathbf{B}\mathbb{Z})$
produces the new transformation
By the rules of horizontal composition of natural transformations, this is the transformation whose component naturality square on $(\bullet \stackrel{1}{\to} \bullet)$ in $\mathbf{B}\mathbb{Z}$ is the diagram
in $\mathbf{B}\mathbb{Z}$, hence the morphism $(g^n \stackrel{g^{\ell} h}{\to} Ad_h g^n)$ in $G//_{Ad}G$. In particular, the categorical circle action is
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.
In Top the notion of free loop space objects reproduces the standard notion of topological free loop spaces.
Let the ambient (∞,1)-category be ∞Grpd, let $G$ be an ordinary group and $\mathbf{B}G$ its one-object delooping groupoid.
We have that the loop groupoid
the action groupoid of the adjoint action of $G$ on itself.
We spell this out in full pedestrian detail, as a little exercise in computing homotopy pullbacks.
We have that the path space object is $\mathbf{B}G^I = [I,\mathbf{B}G]$ – 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
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
in $\mathbf{B}G = {*}//G$.
The pullback of the top right corner of the above defining limit diagram is
identifying the two actions from the right, and then the remaining pullback completing the limit diagram is
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 $(g_1,g_2) \stackrel{h_1,h_2}{\to} (g'_1, g'_2)$ in $G\backslash\backslash (G\times G)//G$ corresponding to a natural transformation
between functors $I\times I \to \mathbf{B}G \times \mathbf{B}G$ may always be decomposed as
Staring at this for a moment shows that this is a unique factorization of every morphism through one of the form
which is naturally identified with a morphism in the action groupoid $G//_{Ad} G$ of the adjoint action of $G$ on itself.
This means that the inclusion
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 $\mathbf{B}G$ of a group is
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… )
The isotropy group of a topos is its free loop space object in the 2-category Topos.
loop space object, free loop space object,
Free loop space objects in the (∞,1)-topos of derived stacks on the site of differential graded algebras are discussed in
More information in the topological case is given in:
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 $\mathcal{L} \mathbf{B}G$.