symmetric monoidal (∞,1)-category of spectra
Hochschild (co)homology is a homological construction which makes sense for any associative algebra, or more generally any dg-algebra or ring spectrum. It has multiple interpretations in higher category theory. Presently, everything below pertains to Hochschild homology of commutative algebras; an exposition of the noncommutative case remains to be written.
Thus, for a commutative ∞-algebra, its Hochschild homology complex is its (∞,1)-tensoring with the ∞-groupoid incarnation of the circle. More generally, for any -groupoid/simplicial set, is the corresponding higher order Hochschild homology of .
In the presence of function algebras on ∞-stacks it may happen that is the algebra of functions on some ∞-stack and that sends powerings of to tensorings of . In that case it follows that the Hochschild homology complex of is the function complex on the derived loop space of .
where and are regarded as -bimodules in the obvious way.
Then it was understood that this complex is the result of tensoring the -bimodules with over but using the derived functor of the tensor product functor – the Tor functor – in the ambient model structure on chain complexes:
Then still a little later, it was understood that this is just the ordinary tensor product in the symmetric monoidal (∞,1)-category of chain complexes. If this is understood, we can just write again simply
This, generally, is the definition of the Hochschild homology object of any bimodule over an monoid in a monoidal (∞,1)-category. Of special interest is the case where . In this case this object is also called the (“-” or “derived-”)center of :
If here can be identified with an -algebra of functions on an object , and if taking functions commutes with -pullbacks, then
is the -algebra of functions on the free loop space object of .
By the Hochschild-Kostant-Rosenberg theorem and its generalizations, the Hochschild homology of an ordinary algebra tends to behave like the algebra of Kähler differentials of . More generally, this computes the cotangent complex of the -algebra . The cup product gives the wedge product of forms and the -action the de Rham differential.
Analogously, Hochschild cohomology of computes the multivector fields on . There are pairing operations on HH homology and cohomology that make them support a general differential calculus on , which makes sense even if is a noncommutative algebra.
We start with the general-abstract definition of Hochschild homology and then look at special and more traditional cases.
We look at the very general abstract definition of Hochschild (co)homology and some important subcases.
We discuss Hochschild homology of commutative algebras for the case that these are related to function algebras on derived loop spaces.
is the Hochschild homology complex of over .
Then for an -perfect object we have
For the circle, this is ordinary Hochschild homology, while for general it is called higher order Hochschild homology .
For the functor that forms symmetric monoidal (∞,1)-categories of quasicoherent ∞-stacks of modules over ∞-stacks over an (∞,1)-site of ∞-algebras for the ordinary theory of commutative -algebras this has setup been considered in detail in (Ben-ZviFrancisNadler).
The following definition formalizes large classes of -perfect objects given by representables.
Let be an (∞,1)-algebraic theory and its (∞,1)-category of -algebras. Let with be a small full sub-(∞,1)-category of which is closed under (∞,1)-limits in and equipped with the structure of a subcanonical (∞,1)-site.
For write for the same object regarded as an object of .
In the context of the above definition we have
This object we call the Hochschild homology complex of .
Generally for higher order Hochschild homology we have
Because the (∞,1)-Yoneda embedding preserves (∞,1)-limits the limit may be computed in . By assumption is closed under limits in . The limit in is the colimit in the opposite (∞,1)-category of -algebras.
This definition of general higher order Hochschild homology by -copowering is
explicit in ToënVezzosi, for ordinary Hochschild homology, hence ,
almost explicit in (GinotTradlerZeinalian), for higher order Hochschild homology for dg-algebras. Details on that are below in the section Higher order Hochschild homology modeled on cdg-algebras
Notice that the tensoring that gives the Hochschild homology is given by the -colimit ove the constant functor
This generalizes to -colimits of functors constant on an algebra, but over a genuine (∞,1)-category diagram.
Specifically let be framed -manifold, an En-algebra and the (∞,1)-category whose objects are framed embeddings of disjoint uniions of open discs into and morphisms are inclusions of these. Let be the functor that assigns to an object corresponding to discs in , and iterated products/units to morphisms
Then the (∞,1)-colimit
is called the topological chiral homology of .
For an ordinary associatve algebra, hence in particular an -algebra, and the circle, this reproduces the ordinary Hochschild homology of (see below).
For more details see (GinotTradlerZeinalian).
We unwind the above generall abstract definition in special classes of examples and find more explicit and more traditional definitions of Hochschild homology.
We demonstrate how the above -category theoretic definition of higher order Hochschild homology reproduces the simplicial definition by (Pirashvili).
it is a simplicial model category;
tensoring with simplicial sets preserves weak equivalences and hence cofibrant replacement.
The -tensoring in an -category presented by a simplicial model category is modeled by the ordinary tensoring of the latter on a cofibrant resolution of the given object. This is discussed in the section ∞-tensoring – models.
We can always use the model structure on homotopy T-algebras to satisfy the assumption of the above proposition. That is a simplicial model category for every and every ordinary algebra is cofibrant in this structure.
Notice that in this model category even if is fibrant (which it is if is an ordinary algebra), then is in general far from being fibant. Computing the simplicial homotopy groups of and hence the Hochschild homology involves passing to a fibrant reolsution of first, that will make it a homotopy T-algebra.
On the other hand, if we find a simplicial model structure on simplicial T-algebras (which are degreewise genuine -algebras) then the coproducts involved degreewise in forming will be tensor products of algebras, and hence in particular themselves again algebras. For such a model the tensoring yields explicitly (under the Dold-Kan correspondence).
This we describe below
for ordinary Hochschild homology in Examples – Simplicial algebra on the circle;
for higher order Hochschild homology of dg-algebras in Higher order Hochschild homology modeled on cdg-algebras.
We first give a detailed discussion of the standard Hochschild complex of a commutative algebra, but from the general abstract -category theoretic point of view.
Then we look in detail at higher order Hochschild homology in the -topos over an (∞,1)-site of formal duals of dg-algebras. In this context the classical theorem by Jones on Hochschild homology and loop space cohomology is a natural consequence of the general machinery.
In derived geometry two categorical gradings interact: a cohesive -groupoid has a space of k-morphisms for all non-negative , and each such has itself a simplicial T-algebra of functions with a component in each non-positive degree. But the directions of the face maps are opposite. We recall the grading situation from function algebras on ∞-stacks.
Functions on a bare -groupoid , modeled as a simplicial set, form a cosimplicial algebra , which under the monoidal Dold-Kan correspondence identifies with a cochain dg-algebra (meaning: with positively graded differential) in non-negative degree
On the other hand, a representable has itself a simplicial T-algebra of functions, which under the monoidal Dold-Kan correspondence also identifies with a cochain dg-algebra, but then necessarily in non-positive degree to match with the above convention. So we write
Taking this together, for a general ∞-stack, its function algebra is generally an unbounded cochain dg-algebra with mixed contributions as above, the simplicial degrees contributing in the positive direction, and the homological resolution degrees in the negative direction:
We consider in detail the classical case of Hochschild (co)homology of an associative algebra approaching it from the general abstract perspective on Hochschild homology.
This section focuses on exposition. The formal context in which the constructions considered here follow from first principles is discussed below in Higher order Hochschild homology modeled on cdg-algebras
We shall use two different equivalent models of the circle in in terms of models in :
the simplicial set
This is not fibrant (not a Kan complex). On the contrary, this is the smallest simplicial model available for the circle, with the least number of horn fillers.
In low degrees it looks as follows
Here for instance the expression denotes the morphism of simplicial sets that sends the first edge (the 2-face) of the 2-simplex to the unique degenerate 1-cell and the second edge (the 0-face) to the unique non-degenerate 1-cell of .
Above in the section on Higher order Hochschild homology we had discussed how the Hochschild homology of is given by the simplicial algebra that is the tensoring of regarded as a constant simplicial algebra with the simplicial set (the 1-simplex with its two 0-cells identified).
We describe now in detail what this simplicial circle algebra looks like. The proof that this construction is indeed homotopy-good is given below in As functions on the derived loop space
When forming the copowering of with the simplicial circle , we get the same structure as displayed above, but with one copy of for each item in parenthesis.
To be very explicit, we recall and demonstrate the following elementary fact.
We check the universal property of the coproduct: for and two morphisms, we need to show that there is a unique morphism such that the diagram
commutes. For the left triangle to commute we need that sends elements of the form to . For the right triangle to commute we need that sends elements of the form to . Since every element of is a product of two elements of this form
this already uniquely determines to be given on elements by the map
That this is indeed an -algebra homomorphism follows from the fact that and are
Notice that for all this it is crucial that we are working with commutative algebras.
We have that the tensoring of with the map of sets from two points to the single point
is the product morphism on . And that the tensoring with the map from the empty set to the point
is the unit morphism on . Generally, for any map of sets we have that the tensoring
is the morphism between tensor powers of of the cardinalities of and , respectively, whose component over a copy of on the right corresponding to is the iterated product on as many tensor powers of as there are elements in the preimage of under .
We see that in low degree the simplicial algebra has the components
The two face maps from degree 1 to degree 0 both come from mapping two points to a single point, so they are both the product on .
The three face maps from degree 3 to degree 2 are more interesting. We have
Notice that for the last one we had to cyclically permute the source in order to display the maps in this planar fashion.
So therefore we get the tensorings
In summary we have so far
The Moore complex of this simplicial algebra is the traditional Hochschild chain complex of
This we describe in more detail in the section Explicit description of the Hochschild complex.
Generally, for any simplicial set, is the simplicial algebra whose Moore complex is the complex that (Pirashvili) uses to define higher order Hochschild homology.
We spell out in detail how in degree 0 and 1 the homology of the Hochschild complex of is that of its Kähler differential forms. Under mild conditions on this is also true in higher degrees, which is the statement of the Hochschild-Kostant-Rosenberg theorem.
The isomorphism is induced by the identifications
where on the left we display elements of under the above identification of these tensor powers in .
By the above discussion, the Moore complex-differential acts on by
The last term on the right is precisely the term by which one has to quotient out the module of formal expressions to get the module of Kähler differentials: setting it to 0 is the derivation property of
Therefore we have manifestly
We may also compute the -homology on the normalized chain complex, which is in degree 1 the quotient of by the image of the degeneracy map , which is
and thus maps
So passage to the normalized chains imposes the condition
where for and we have
For instance for we have
and the tensor product (in !) is componentwise
The automorphism 2-group of the categorical circle is
We may compute the automorphism 2-group in the full sub-(∞,1)-category Grpd ∞Grpd, whose morphisms are functors and 2-morphisms are natural isomorphisms (see the statement about homotopy 1-types at homotopy hypothesis for details). A functor between delooping groupoids is precisely a group homomorphism . The additive group endomorphisms of are precisely given by multiplication with elements in , the two automorphisms in there are .
The natural transformations between such functors are
Now consider the right homotopy that exhibits the morphism 1 in .
This means that under copowering this on
we get in degree 0 the morphism
Under the above identification of the homology of with Kähler forms, this is on elements the map
(automorphisms of the odd line)
This means that under the identification of with functions on the odd line,in degree 0 this corresponds to the even vector field on the odd line, and in degree 1 to the odd vector field .
regarded as a chain complex in -bimodules for the evident bimodule structure in each degree.
The Hochschild homology of with coefficients in is the homology of the Hochschild chain complex, written
At the level of the underlying -modules we have natural isomorphisms
given on elements by sending
The action of the differential in on elements of the latter form is then
In words this means that the Hochschild complex is obtained froms the bar complex by “gluing the two ends of a sequence of elements of to a circle by a bimodule”.
The fact that the circle appears here has in fact a deep significance: the Hochschild chain complex may be understood in higher geometry as encoding functions on a free loop space object of whatever behaves like being functions on.
The (∞,1)-category of ∞-algebras over is presented by the model structure on simplicial commutative k-algebras .
be the (∞,1)-sheaf (∞,1)-topos over .
For an ordinary -algebra, we say that the free loop space object
of formed in is the derived loop space of .
The term derived is just to emphasize that we do not form the free loop space object in an (∞,1)-topos of -sheaves over a 1-site inside the 1-category . These “underived” (not embedded into (∞,1)-category theory) free loop space objects would just be equivalent to . The derived loop space instead has rich interesting structure.
Since ∞-stackification is a left exact (∞,1)-functor and hence preserves finite (∞,1)-limits, we have that the defining pullback for may be computed in the (∞,1)-category of (∞,1)-presheaves . Since the (∞,1)-Yoneda embedding preserves all (∞,1)-limits this in turn may be computed in the (∞,1)-site , hence by assumption in . The relevant -pullback there is the claimed -pushout in the opposite (∞,1)-category .
The -algebra of functions on the derived loop space of is when modeled by a simplicial algebra in under the monoidal Dold-Kan correspondence equivalent to the Hochschild chain complex of with coefficients in itself:
By the discussion at homotopy T-algebra we may model by the injective model structure on simplicial presheaves on , left Bousfield localized at the morphisms . This localized model structure we write .
where both morphism are simple the product on . By general properties of homotopy pushouts and the injective model structure on simplicial presheaves we have that this homotopy pushout is computed by an ordinary pushout once we pass to a weakly equivalent diagram in which one of the two morphism is a cofibration of simplicial algebras.
It is sufficient to find a resolution in the global model structure because left Bousfield localization strictly increases the class of weak equivalences, so that every gloabl weak equivalence is also a local weak equivalence.
Since we are in the injective model structure this just means that this morphism needs to be over each in a monomorphism of simplicial sets. If we find also as a strictly product-preserving functor (notice that the general functor in our model category need not even preserve products weakly, it will do so after fibrant replacement) then it being monomorphism over implies that it is monic over every .
There is a standard resolution of the kind we need called the bar complex, see for intance (Ginzburg, page 16) for an explicit description. This is usually discussed as a chain complex in the category of -modules. But in fact after applying the Dold-Kan correspondence to regard it as a simplicial module it is naturally even a simplicial object in :
with the evident face and degeneracy maps given by binary product operation in the algebra and insertion of units.
Take the morphism degreewise to be the inclusion of as the two outer direct summands
where is the monoid unit.
This is clearly degreewise a monomorphism, hence is a monomorphism. Under the Moore complex functor it maps to the standard bar complex resolution as found in the traditional literature (as reviewed for instance in Ginzburg). This morphism of chain complexes is an isomorphism in homology. Since under the Dold-Kan correspondence simplicial homotopy groups are identified with homology groups, we find that indeed is a weak equivalence in and hence in .
We may now compute the pushout in and this will compute the desired homotopy pushout. Notice that this pushout indeed takes place just in simplicial copresheaves, not in product-preserving copresheaves!
But this ordinary pushout it manifestly the claimed one.
crucially uses the assumption that is a commutative algebra;
curiously does not make use of any specific property of the set of morphisms at which we are considering the left Bousfield localization. The entire construction proceeds entirely at the underlying simplicial sets of our simplicial algebras. In fact, the resulting homotopy pushout is a simplicial copresheaf on that no longer preserves any products: there is no manifest algebra structure.
But also, this object is far from being fibant in the localized model structure . The Bousfield localization, hence the information about the set of maps at which we are localizing, hence the algebra structure, kicks in only once we pass now to the fibrant resolution of our pushout. That fibrant replacement equips the Hochschild chain complex with the structure of an -algebra.
We discuss details of Hochschild homology in the context of dg-geometry: the (∞,1)-topos over an (∞,1)-site of formal duals of commutative dg-algebras over a field, presented by the model structure on dg-algebras.
For the discussion of Hochschild homology in this , the main fact about the model structure on dg-algebras that we need is this:
In the projective model structure on unbounded commutative dg-algebras over we have that
By the above fact Pirashvili’s copowering definition of higher order Hochschild homology holds true in dg-geometry. For a manifold regarded as a topological space and then as a constant ∞-stack in we have for any
Jones’ theorem asserts that the Hochschild homology of the dg-algebra of differential forms on a manifold computes the homology of the corresponding loop space. We discuss now how this result follows using derived loop spaces of constant ∞-stacks.
We sketch the proof in terms of the above derived loop space technology.
Since is -perfect (…) we have by the above copowering-description of the Hochschild complexes that the cohomology of the loop space of
is given by the Hochschild complex of the dg-algebra
Consider as before the categorical circle as the corresponding constant ∞-stack in . We describe the function -algebra on . Below this will serve to explain the nature of the canonical circle action on the Hochschild complex of a cdg-algebra.
We have an equivalence
in . The (∞,1)-functor is also left adjoint, so that
in . Since the point is representable, we have by the definition of as the left -Kan extension of the inclusion that this is
This is the formula for the -power of the cdg-algebra by by -groupoid . By the above fact, using that the circle is a finite -groupoid, this is given by the cdg-algebra of polynomial differential forms on simplices of
This means that is no longer the circle itself, but the odd line, regarded with its canonical -grading.
This point is amplified in (Ben-ZivNadler).
We have that
There is rich algebraic structure on Hochschild homology and cohomology itself, and on the pairing of the to. We describe various aspects of this.
It turns out that
Hochschild homology of encodes Kähler differential forms on ;
Hochschild cohomology of encodes multivector fields on ;
there are natural pairings between and that mimic the structure of the natural pairings between vector fields and differential forms on smooth manifold.
The Hochschild-Kostant-Rosenberg theorem states that under suitable conditions, the Hochschild homology of an algebra (with coefficients in itself) computes the wedge powers of its Kähler differentials.
of the first Hochschild homology of with coefficients in itself and degree-1 Kähler differential forms of .
For Write for the -fold wedge product of with itself: the degree -Kähler-differentials.
The isomorphism extends to a graded ring morphism
If the -algebra is sufficiently well-behaved, then this morphism is an isomorphism that identifies the Hochschild homology of in degree with for all :
essentially of finite type
then there is an isomorphism of graded -algebras
Moreover, dually, there is an isomorphism of Hochschild cohomology with wedge products of derivations:
The next statement is known as the Deligne conjecture.
The higher order Hochschild homology of an object with respect to the -sphere and with coefficients in a geometric function object is naturally an E(d+1)-algebra): an algebra over an operad over the little k-cubes operad for .
in . Then the integral transforms on sheaves
induced by these spans constitute the -action on the function objects on .
This was observed in (Ben-ZviFrancisNadler, corollary 6.8).
Some historical comments on the Deligne conjecture.
Historically it was first found that there is the structure of a Gerstenhaber algebra on . By (Cohen) it was known that Gerstenhaber algebras arise as the homology of E2-algebras in chain complexs. In a letter in 1993 Deligne wondered whether the Gerstenhaber structure on the Hochschild cohomology lifts to an E2-algebra-structure on the cochain complex .
In GerstenhaberVoronov (1994) a resolution of the Gerstenhaber algebra structure was given, but the relationship to -algebras remained unclear.
In (Tamarkin (1998)) a genuine resolution in the model structure on operads of the Gerstenhaber operad was given and shown to act via the Gerstenhaber-Voronov construction on . This proved Deligne’s conjecture.
Various authors later further refined this result. A summary of this history can be found in (Hess).
Notice that the identification of Hochschild (co)homology as coming from higher order free loop spaces makes all this structure manifest.
The function algebra on is the cosimplicial algebra of singular cochains on . Under the monoidal Dold-Kan correspondence it identifies with the cochain dg-algebra that computes the singular cohomology of .
Apply the central identification . Then observe that the free loop space object of the constant -stack is the constant -stack on the ordinary free loop space, because is a left exact (∞,1)-functor and because in Top. Then use by the above remark that is singular cochains on .
This result, which follows directly from the general abstract desciption of Hichschild homology is known as Jones’ theorem. We now review the results in the literature on this point.
Let be a compact manifold oriented smooth manifold of dimension . Write for the dg-algebra of cochains for singular cohomology of . Write for the topological free loop space of and for its singular homology.
There is a linear isomorphism of degree
This is due to (FelixThomasVigue-Poirrier, section 7)).
There is an isomorphism
This is due to (Jones).
This is (Menichi, theorem 3).
Two extensions are equivalent if there is an isomorphism or -algebra that makes
Due to the -splitness assumption there is an isomorphism of -modules and this is equipped with a -algebra structure such that the product on the direct summand is that of . From this we find that the product on is of the form
where is some -linear map. Since the product on is (by definition) associative, it follows that for that this satisfies for all the cocycle equation
as an equation in . This says that must be a Hochschild cocycle
Conversely, every such cocycle yields a -split extension of by this way:
For a -algebra and an -bimodule, equivalence classes of Hochschild extensions of by are in bijection with degree 2 Hochschild cohomology .
See for instance Weibel, theorem 9.3.1.
As a special case of the above statement about extensions of , we obtain a statement about deformation of .
A standard problem is to deform a -algebra by introducing a new “parameter” that squares to 0 – and a new product
From the above we see that this is the same as finding an -split extension of by itself. So in particular such extensions are given by Hochschild cocycles .
Hochschild cohomology of ordinary algebras was introduced in
A textbook discussion is for instance in chapter 9 of
or in chapter 4 of
The definition of the higher order Hochschild complex as (implicitly) the tensoring of an algebra with a simplicial set is due to
A survey of traditional higher order Hochschild (co)homology and further developments and results are described in
A considerably refined discussion of this which almost makes the construction of Hochschild complexes as an -copowering operation manifest is in
Specifically the dicussion of differential forms via such an -category theoretic perspective of the HKR-theorem is discussed in
General homotopy-theoretic setups and results for contexts in which this makes sense are discussed in
Jones’s theorem is due to
The BV-algebra structure on Hochschild cohomology of singular cochain algebras is discussed in
The abstract differential caclulus on is discussed for instance in
A review of Deligne’s conjecture and its solutions is in
More developments are in
Relation to factorization homology is discussed in
For more references on the relation to topological chiral homology see there.
Interesting wishlists for treatments of Hochschild cohomology are in this MO discussion.