Stabe homotopy theory
Stable homotopy theory
In an (∞,1)-category with -pullbacks, the free loop space object of any object – also called the inertia groupoid – is an object that behaves as if its generalized elements are loops in , morphisms between generalized elements homotopies of loops, and so on.
For the case that Top this reproduces the ordinary notion of free loop space objects of topological spaces.
Over each fixed element , the free loop space object looks like the based loop space object of .
Free loop space objects come naturally equipped with various structures of interest, such as a categorical circle action. The cohomology of is Hochschild cohomology or cyclic cohomology of function algebras on . The categorical circle action induces differentials on these cohomolgies, identifying them, in suitable cases, with algebras of Kähler differential forms on .
In higher category theory
In an (∞,1)-category with (∞,1)-pullbacks, for an object, its free loop space object is the -pullback of the diagonal along itself
In homotopy type theory
In the literature (see below) the free loop space object is sometimes described heuristically as: “a point of is a choice of making two points of 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 ;
of the product type .
See the discussion at homotopy pullback in the section Construction in homotopy type theory for how this is equivalent to the previous definition.
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 -pullback as the ordinary limit
where is a path space object for . At least if we have the structure of a model category we may take for a path space object of .
From this description one sees that is built from pairs of paths in 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 over a point is the corresponding (based) loop space object of : we have an -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 from two consecutive paths, the first starting at the basepoint , the second ending there. This is weakly equivalent to the based loop space object built from just the path space object with a single copy of , by standard arguments as for instance form page 12 on in
In an -topos
We consider now the case that is an (∞,1)-topos (of (∞,1)-sheaves/∞-stacks). This comes canonically with its terminal global sections (∞,1)-geometric morphism
As a mapping space object
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 we still write for the constant ∞-stack on this, the image of this under . Since is a left adjoint and hence preserves this poushout, there is no risk of confusion.
To see that the given -pushout indeed produces the circle, we use the standard model structure on simplicial sets to present ∞Grpd. In the -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
There is a natural equivalence
This follows from the above by the fact (see closed monoidal structure on (∞,1)-toposes) that the internal hom in an -topos preserves finite colimits in its first argument and satisfies
We have that the free loop space object of is equivalently the powering of by the -groupoid :
Follows by the above from the equivalence discussed at (∞,1)-topos.
Intrinsic circle action
By precomposition, the automorphism 2-group of the circle acts on free loop space of an object
The connected component of on the identity is equivalent to
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, in mind.
We may realize the circle Top under as the [delooping]] groupoid of the additive group of integers
The automorphism 2-group of this object is the functor groupoid
whose objects are invertible functors and whose morphisms are natural transformations between these.
The functors correspond bijectively to group homomorphisms , hence to multiplication by
Natural transformations between two such endomorphisms are given by a component such that all diagrams
commute in . This can happen only for , but then it happens for arbitrary .
In other words we have
The object corresponds to the self-mapping of the circle that fixes the basepoint and has winding number . The transformation corresponds then to a rigid rotation of the loop by full circles
Notably for and 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 -times around the circle (the rest of the diagram). Of course since we are using a model of 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 of the standard circle. The is the groupoid with 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 is the skeleton of .
Consider ∞Grpd, a group and the delooping groupoid?. Then (as discussed in detail below). A morphism in corresponds to a natural transformation
Precomposing this with the automorphism of the object in
produces the new transformation
By the rules of horizontal composition of natural transformations, this is the transformation whose component naturality square on in is the diagram
in , hence the morphism in . In particular, the categorical circle action is
Hochschild cohomology and cyclic cohomology
quasicoherent ∞-stacks on form the Hochschild homology object of (if the axioms of geometric function theory are met) as described there. The circle acton on induces differentials on these.
… details to be written, but see Hochschild cohomology and cyclic cohomology for more.
Free topological loop spaces
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 be an ordinary group and its one-object delooping groupoid.
We have that the loop groupoid
the action groupoid of the adjoint action of 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 – the functor groupoid, where is the free groupoid on the standard interval object – which is (by the definition of natural transformation) the action groupoid
for the action of on itself, by inverse left and direct right multiplication separately: the naturality square of a natural transformation defining a morphism in this groupoid is the commuting square
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 is the action groupoid of acting diagonally on 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 in corresponding to a natural transformation
between functors 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 of the adjoint action of 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 of a group is
We describe how the Chern character of vector bundles over may be realized in terms of the cohomology of the free loop space object .
Assume now is a nice category of smooth spaces, and let be an object of .
Consider a group object in and a representation of given my a group homomorphism to the general linear group (in ): . For instance could be itself and this morphism the identity.
The trace of the representation is invariant under conjugation in the group and so defnes a map – a class function. By the equivalence discussed above, this may be regarded as a characteristic class
on the free loop space of .
The cocycle of a -principal bundle on 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 -invariance of the trace is now -invariant and hence defines an element in the cyclic cohomology of .
The Hom-space is a model for the graded commutative algebra of complex-valued differential forms on , with the categorical circle action corresponding to the de Rham differential. Hence is a model for closed forms and maps to de Rham cohomology of . If the de Rham theorem holds for in , then this may be identified with the real cohomology .
In the case that , the compatibility of the trace with direct sums and tensor products of vector bundles over makes the above construction a ring homomorphism from the topological K-theory of to de Rham cohomology, hence a very good candidate to being the Chern character
( to be completed… )
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 for a space which is the classifying space of a crossed module of groups. This generalises the above example of .