group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 $A$ a commutative ∞-algebra, its Hochschild homology complex is its (∞,1)-tensoring $S^1 \cdot A$ with the ∞-groupoid incarnation of the circle. More generally, for $S$ any $\infty$-groupoid/simplicial set, $S \cdot A$ is the corresponding higher order Hochschild homology of $X$.
In the presence of function algebras on ∞-stacks it may happen that $A = \mathcal{O}(X)$ is the algebra of functions on some ∞-stack $X$ and that $\mathcal{O}(-)$ sends powerings of $X$ to tensorings of $\mathcal{O}(X)$. In that case it follows that the Hochschild homology complex of $\mathcal{O}(X)$ is the function complex $\mathcal{O}(\mathcal{L}(X))$ on the derived loop space $\mathcal{L}X$ of $X$.
Originally the notion of Hochschild homology was introduced as the chain homology of a certain chain complex associated to any bimodule $N$ over some algebra $A$: its bar complex, written
where $N$ and $A$ are regarded as $A \otimes A^{op}$-bimodules in the obvious way.
Then it was understood that this complex is the result of tensoring the $A$-bimodules $N$ with $A$ over $A \otimes A^{op}$ 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 $N = A$. In this case this object is also called the (“$(\infty,1)$-” or “derived-”)center of $A$:
If here $A = \mathcal{O}(X)$ can be identified with an $\infty$-algebra of functions on an object $X$, and if taking functions commutes with $(\infty,1)$-pullbacks, then
is the $\infty$-algebra of functions on the free loop space object of $X$.
By the Hochschild-Kostant-Rosenberg theorem and its generalizations, the Hochschild homology $HH_\bullet(\mathcal{O}(X),\mathcal{O}(X))$ of an ordinary algebra tends to behave like the algebra of Kähler differentials of $\mathcal{O}(X)$. More generally, this computes the cotangent complex of the $\infty$-algebra $\mathcal{O}(X)$. The cup product gives the wedge product of forms and the $S^1$-action the de Rham differential.
Dually this means that in derived geometry the free loop space object $\mathcal{L} X$ consists of infinitesimal loops in $X$ (in ordinary geometry it would be equal to $Spec A$, consisting only of constant loops).
Analogously, Hochschild cohomology $HH^\bullet(\mathcal{O}(X), \mathcal{O}(X))$ of $\mathcal{O}(X)$ computes the multivector fields on $X$. There are pairing operations on HH homology and cohomology that make them support a general differential calculus on $X$, which makes sense even if $\mathcal{O}(X)$ 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.
Let $\mathbf{H}$ be an (∞,1)-topos that admits function algebras on ∞-stacks (see there for details)
In particular the (∞,1)-category of ∞-algebras $Alg^{op}$ is (∞,1)-tensored over ∞Grpd. Then for $A \in Alg$ and $K \in \infty Grpd$ we say that
is the Hochschild homology complex of $A$ over $K$.
We say a full sub-(∞,1)-category of $\mathbf{H}$ consists of $\mathcal{O}$-perfect objects if on these $\mathcal{O}$ commutes with (∞,1)-limits.
Then for $X$ an $\mathcal{O}$-perfect object we have
For $K = S^1$ the circle, this is ordinary Hochschild homology, while for general $K$ it is called higher order Hochschild homology .
For $\mathcal{O}$ 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 $k$-algebras this has setup been considered in detail in (Ben-ZviFrancisNadler).
The following definition formalizes large classes of $\mathcal{O}$-perfect objects given by representables.
Let $T$ be an (∞,1)-algebraic theory and $T Alg_\infty$ its (∞,1)-category of $\infty$-algebras. Let $C$ with $T \hookrightarrow C \hookrightarrow T Alg_\infty^{op}$ be a small full sub-(∞,1)-category of $T Alg_\infty^{op}$ which is closed under (∞,1)-limits in $T Alg$ and equipped with the structure of a subcanonical (∞,1)-site.
Take $\mathbf{H} := Sh(C)$ the (∞,1)-category of (∞,1)-sheaves on $C$. This is an (∞,1)-topos for derived geometry modeled on $T Alg_\infty$. Write $C \hookrightarrow \mathbf{H}$ for the (∞,1)-Yoneda embedding.
For $X \in C\stackrel{}{\hookrightarrow} \mathbf{H}$ write $\mathcal{O}(X)$ for the same object regarded as an object of $T Alg_\infty$.
In the context of the above definition we have
where on the right we have the (∞,1)-tensoring of $T Alg_\infty$ over $\infty Grpd$, which is the (∞,1)-colimit over the diagram $S^1$ of the (∞,1)-functor constant on $\mathcal{O}(X)$
This object we call the Hochschild homology complex of $\mathcal{O}X$.
Generally for higher order Hochschild homology we have
Because the (∞,1)-Yoneda embedding preserves (∞,1)-limits the limit $X^K$ may be computed in $C$. By assumption $C$ is closed under limits in $T Alg_\infty^{op}$. The limit $X^K$ in $T Alg^{op}$ is the colimit $K \cdot \mathcal{O}(X)$ in the opposite (∞,1)-category of $\infty$-algebras.
This definition of general higher order Hochschild homology by $(\infty,1)$-copowering is
explicit in ToënVezzosi, for ordinary Hochschild homology, hence $K = S^1$,
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
explicit in Ben-Zvi/Francis/Nadler, corollary 4.12 for HH with values in quasicoherent ∞-stacks and over perfect $\infty$-stacks (see there for details).
Notice that the tensoring that gives the Hochschild homology is given by the $\infty$-colimit ove the constant functor
This generalizes to $\infty$-colimits of functors constant on an algebra, but over a genuine (∞,1)-category diagram.
Specifically let $X$ be framed $n$-manifold, $A$ an En-algebra and $D_X$ the (∞,1)-category whose objects are framed embeddings of disjoint uniions of open discs into $X$ and morphisms are inclusions of these. Let $F_A$ be the functor that assigns $A^{k}$ to an object corresponding to $k$ discs in $X$, and iterated products/units to morphisms
Then the (∞,1)-colimit
is called the topological chiral homology of $X$.
For $A$ an ordinary associatve algebra, hence in particular an $E_1$-algebra, and $X$ the circle, this reproduces the ordinary Hochschild homology of $A$ (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 $(\infty,1)$-category theoretic definition of higher order Hochschild homology reproduces the simplicial definition by (Pirashvili).
Let $T$ be a Lawvere theory regarded as a 0-truncated (∞,1)-algebraic theory.
Consider a model structure on simplicial T-algebras/on homotopy T-algebras presenting $T Alg_\infty$ such that
it is a simplicial model category;
tensoring with simplicial sets preserves weak equivalences and hence cofibrant replacement.
Then for $\mathcal{O}(X) \in T Alg \hookrightarrow T Alg_\infty$ and $K \in \infty Grpd$ the higher order Hochschild homology complex $K \cdot \mathcal{O}(X)$ is presented by the ordinary tensoring $K \cdot \mathcal{O}(X)$ in the model category, for $K$ any simplicial set incarnation of the $\infty$-groupoid.
The $(\infty,1)$-tensoring in an $(\infty,1)$-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 $T$ and every ordinary algebra is cofibrant in this structure.
Notice that in this model category even if $\mathcal{O}(X)$ is fibrant (which it is if $\mathcal{O}X$ is an ordinary algebra), then $K \cdot \mathcal{O}(X)$ is in general far from being fibant. Computing the simplicial homotopy groups of $K \cdot \mathcal{O}(X)$ and hence the Hochschild homology involves passing to a fibrant reolsution of $K \cdot \mathcal{O}(X)$ 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 $T$-algebras) then the coproducts involved degreewise in forming $K \cdot \mathcal{O}(X)$ will be tensor products of algebras, and hence in particular themselves again algebras. For such a model the tensoring $K \cdot \mathcal{O}(X)$ yields explicitly (under the Dold-Kan correspondence).
This is the case for the tensoring of dg-algebras over simplicial sets and leads to Teimuraz Pirashvili’s formulation of higher order Hochschild homology for ordinary algebras (Pirashvili).
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 $(\infty,1)$-category theoretic point of view.
Then we look in detail at higher order Hochschild homology in the $(\infty,1)$-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 $\infty$-groupoid $X$ has a space of k-morphisms $X_k$ for all non-negative $k$, 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 $\infty$-groupoid $K$, modeled as a simplicial set, form a cosimplicial algebra $\mathcal{O}(K)$, 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 $X$ 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 $X_\bullet$ 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 $\infty Grpd$ in terms of models in $sSet$:
the simplicial set $\Delta[1]/\partial \Delta[1]$
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 $(* * \to * )$ denotes the morphism of simplicial sets $\Delta[2] \to \Delta[1]/\partial \Delta[1]$ 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 $\Delta[1]/ \partial \Delta[1]$.
the nerve of the delooping groupoid $\mathbf{B}\mathbb{Z}$ of the additive group of integers.
This model is fibrant (is a Kan complex) and makes manifest the group structure on $S^1$, which is the strict 2-group structure on $\mathbf{B}\mathbb{Z}$ or equivalently the structure of a simplicial group on its nerve.
Let $A \in CAlg_k$ be a commutative associative algebra over a commutative ring $k$.
Above in the section on Higher order Hochschild homology we had discussed how the Hochschild homology of $A$ is given by the simplicial algebra $(\Delta[1]/\partial \Delta[1]) \cdot A \in CAlg_k^{\Delta^{op}}$ that is the tensoring of $A$ regarded as a constant simplicial algebra with the simplicial set $\Delta[1]/\partial \Delta[1]$ (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 $A$ with the simplicial circle $S^1$, we get the same structure as displayed above, but with one copy of $A$ for each item in parenthesis.
To be very explicit, we recall and demonstrate the following elementary fact.
In $CAlg_k$ the coproduct is given by the tensor product over $k$:
We check the universal property of the coproduct: for $C \in CAlg_k$ and $f,g : A,B \to C$ two morphisms, we need to show that there is a unique morphism $(f,g) : A \otimes_k B \to C$ such that the diagram
commutes. For the left triangle to commute we need that $(f,g)$ sends elements of the form $(a,e_B)$ to $f(a)$. For the right triangle to commute we need that $(f,g)$ sends elements of the form $(e_A, b)$ to $g(b)$. Since every element of $A \otimes_k B$ is a product of two elements of this form
this already uniquely determines $(f,g)$ to be given on elements by the map
That this is indeed an $k$-algebra homomorphism follows from the fact that $f$ and $g$ are
Notice that for all this it is crucial that we are working with commutative algebras.
We have that the tensoring of $A$ with the map of sets from two points to the single point
is the product morphism on $A$. And that the tensoring with the map from the empty set to the point
is the unit morphism on $A$. Generally, for $f : S \to T$ any map of sets we have that the tensoring
is the morphism between tensor powers of $A$ of the cardinalities of $S$ and $T$, respectively, whose component over a copy of $A$ on the right corresponding to $t \in T$ is the iterated product $A^{\otimes_k |f^{-1}\{t\}|} \to A$ on as many tensor powers of $A$ as there are elements in the preimage of $t$ under $f$.
We see that in low degree the simplicial algebra $(\Delta[1]/\partial \Delta[1]) \cdot A$ 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 $A$.
The three face maps from degree 3 to degree 2 are more interesting. We have
and
and
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
and
and
In summary we have so far
The Moore complex of this simplicial algebra is the traditional Hochschild chain complex of $A$
This we describe in more detail in the section Explicit description of the Hochschild complex.
Generally, for $K$ any simplicial set, $K \cdot A$ 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 $A$ is that of its Kähler differential forms. Under mild conditions on $A$ this is also true in higher degrees, which is the statement of the Hochschild-Kostant-Rosenberg theorem.
The homology of the Hochschild complex $S^1 \cdot A$ in degree 1 is the Kähler differential forms of $A$
The isomorphism is induced by the identifications
where on the left we display elements of $A^{\otimes_k}$ under the above identification of these tensor powers in $S^1 \cdot A$.
By the above discussion, the Moore complex-differential acts on $(f,g,h) \in A \otimes_k A \otimes_k A$ by
The last term on the right is precisely the term by which one has to quotient out the module of formal expressions $f \wedge d g$ to get the module of Kähler differentials: setting it to 0 is the derivation property of $d$
Therefore we have manifestly
We may also compute the $\partial$-homology on the normalized chain complex, which is in degree 1 the quotient of $A \otimes_k A$ by the image of the degeneracy map $\sigma : A \to A \otimes_k A$, which is
and thus maps
So passage to the normalized chains imposes the condition
Under the identification of $HH_\bullet(A,A) = H_\bullet(S^1 \cdot A)$ with Kähler differential forms, the cup product on homology identifies with the wedge product of differential 0- and 1-forms.
Under the monoidal Dold-Kan correspondence the product on the Moore complex $N_\bullet(S^1 \cdot A)$ is given by the Eilenberg-Zilber map $\nabla$
where for $\omega \in (S^1\cdot A)_p$ and $\lambda \in (S^1 \cdot A)_q$ we have
For instance for $\omega = f \wedge d g \in (S^1 \cdot A )_0$ we have
and
and the tensor product (in $(S^1 \cdot A)_2$!) is componentwise
Therefore
We describe the canonical action of the automorphism 2-group of the circle $S^1$ on $S^1 \cdot A$ and how its degree-1 part induces under the above identification $H_\bullet(S^1 \cdot A) \simeq \Omega^\bullet_K(A)$ the action of the de Rham differential.
The automorphism 2-group of the categorical circle is
We may compute the automorphism 2-group in the full sub-(∞,1)-category Grpd $\subset$ ∞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 $\mathbf{B}G \to \mathbf{B}H$ is precisely a group homomorphism $G \to H$. The additive group endomorphisms of $\mathbb{Z}$ are precisely given by multiplication with elements in $\mathbb{Z}$, the two automorphisms in there are $\pm 1$.
The natural transformations between such functors are
Now consider the right homotopy that exhibits the morphism 1 in $Aut(\mathbf{B}\mathbb{Z})_{Id}$.
This sends
This means that under copowering this on $A$
we get in degree 0 the morphism
Under the above identification of the homology of $\mathbf{B}\mathbb{Z} \cdot A$ with Kähler forms, this is on elements the map
(automorphisms of the odd line)
This means that under the identification of $(\mathbf{B}\mathbb{Z}) \cdot k \simeq C^\infty(k^{0|1})$ with functions on the odd line,in degree 0 this corresponds to the even vector field $\theta \partial/\partial \theta$ on the odd line, and in degree 1 to the odd vector field $\partial/\partial\that$.
(…)
We spell out explicitly the Hochschild chain complex for an associative algebra (over some ring $k$) with coefficients in a bimodule.
The bar complex of $A$ is the connective chain complex
which in degree $n$ has the $(n+1)$ tensor power of $A$ with itself, and whose differential is given by
regarded as a chain complex in $A$-bimodules for the evident bimodule structure in each degree.
Let $N$ be an $A$-bimodule. The Hochschild chain complex $C_\bullet(A,N)$ of $A$ with coefficients in $N$ is the chain complex obtained by taking in the bar complex degreewise the tensor product of $A$-bimodules with $N$:
The Hochschild homology of $A$ with coefficients in $N$ is the homology of the Hochschild chain complex, written
At the level of the underlying $k$-modules we have natural isomorphisms
given on elements by sending
The action of the differential in $C_\bullet(A,N)$ 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 $A$ 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 $A$ behaves like being functions on.
We give a formal derivation of the Hochschild complex of an ordinary commutative associative algebra $\mathcal{O}(X)$ as the function algebra on the derived loop space object $\mathcal{L}X$ in the context of derived geometry.
So let now $T$ be the Lawvere theory of ordiary commutative associative algebras over a field $k$, regard as a 0-truncated (∞,1)-algebraic theory.
The (∞,1)-category $CAlg(k)_\infty$ of ∞-algebras over $T$ is presented by the model structure on simplicial commutative k-algebras $(CAlg_k^{\Delta^{op}})_{proj}$.
This is Quillen equivalent to the standard model structure on connected dg-chain algebras.
The first statement is discussed at (∞,1)-algebraic theory and homotopy T-algebra. The second statement is discussed at monoidal Dold-Kan correspondence.
Let
be a subcanonical (∞,1)-site that is a full sub-(∞,1)-category of formal duals of $\infty$-$T$-algebras, closed under (∞,1)-limits in $T Alg_\infty^{op}$.
Let
be the (∞,1)-sheaf (∞,1)-topos over $C$.
Following the notation at Isbell duality and function algebras on ∞-stacks we write $\mathcal{O}(X) \in T Alg_\infty$ for an object that under the (∞,1)-Yoneda embedding $C \hookrightarrow T Alg_\infty^{op} \to \mathbf{H}$ maps to an object called $X$ in $\mathbf{H}$.
For $\mathcal{O}(X) \in T Alg \hookrightarrow T Alg_\infty$ an ordinary $T$-algebra, we say that the free loop space object
of $X$ formed in $\mathbf{H}$ is the derived loop space of $X$.
The term derived is just to emphasize that we do not form the free loop space object in an (∞,1)-topos of $(\infty,1)$-sheaves over a 1-site inside the 1-category $Alg_k^{op}$. These “underived” (not embedded into (∞,1)-category theory) free loop space objects would just be equivalent to $X$. The derived loop space instead has rich interesting structure.
But if the ambient context of higher geometry over the genuine (∞,1)-site of formal duals to $\infty$-algebras is clear, we can just speakk of free loop space objects . They are canonically given.
We have that $\mathcal{O}(\mathcal{L}X)$ is given by the (∞,1)-pushout in $CAlg_\infty$
hence by the universal cocone
Since ∞-stackification $L : PSh_{(\infty,1)}(C) \to \mathbf{H}$ is a left exact (∞,1)-functor and hence preserves finite (∞,1)-limits, we have that the defining pullback for $\mathcal{L}X$ may be computed in the (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(C)$. Since the (∞,1)-Yoneda embedding preserves all (∞,1)-limits this in turn may be computed in the (∞,1)-site $C$, hence by assumption in $T Alg_\infty$. The relevant $(\infty,1)$-pullback there is the claimed $(\infty,1)$-pushout in the opposite (∞,1)-category $T Alg_\infty$.
The $\infty$-algebra $\mathcal{O} \mathcal{L}X$ of functions on the derived loop space of $X$ is when modeled by a simplicial algebra in $CAlg_k^{\Delta^{op}}$ under the monoidal Dold-Kan correspondence equivalent to the Hochschild chain complex of $\mathcal{O}X$ with coefficients in itself:
First observe that the coproduct in $CAlg_k$ is the tensor product of commutative algebras over $k$
By the discussion at homotopy T-algebra we may model $T Alg_\infty$ by the injective model structure on simplicial presheaves on $T^{op}$, left Bousfield localized at the morphisms $T[k] \otimes T[l] \to T[k+l]$. This localized model structure we write $[T, sSet]_{inj,prod}$.
By the above proposition we have that $\mathcal{O}\mathcal{L}X$ is given by the homotopy pushout in $[T, sSet]_{inj,prod}$ of
where both morphism are simple the product on $\mathcal{O}(X) \in CAlg_k$. 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 $\mathrm{B} \mathcal{O}(X)$ in the global model structure $[T, sSet]_{inj}$ 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 $\mathcal{O}(X) \otimes_k \mathcal{O}(X) \to \mathrm{B} \mathcal{O}X$ needs to be over each $x^n$ in $T$ a monomorphism of simplicial sets. If we find $\mathrm{B} \mathcal{O}X$ 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 $x^1$ implies that it is monic over every $x^n$.
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 $\mathcal{O}(X)$-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 $CAlg_k$:
with the evident face and degeneracy maps given by binary product operation in the algebra and insertion of units.
Take the morphism $\mathcal{O}(X) \otimes \mathcal{O}(X) \to \mathrm{B} \mathcal{O}(X)$ degreewise to be the inclusion of $\mathcal{O}(X) \otimes \mathcal{O}(X)$ as the two outer direct summands
where $e : k \to \mathcal{O}(X)$ is the monoid unit.
This is clearly degreewise a monomorphism, hence is a monomorphism. Under the Moore complex functor $N : Ab^{\Delta^{op}} \to Ch_\bullet^+$ 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 $\mu : \mathrm{B}\mathcal{O}(X) \to \mathcal{O}(X)$ is a weak equivalence in $[T,sSet]_{inj}$ and hence in $[T, sSet]_{inj,prod}$.
We may now compute the pushout in $[T, sSet]$ 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.
This derivation
crucially uses the assumption that $A$ is a commutative algebra;
curiously does not make use of any specific property of the set of morphisms $\{T[k] \coprod T[l] \to T[k+l] \}$ 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 $\mathcal{O}(X) \coprod_{\mathcal{O}(X) \otimes \mathcal{O}(X)} \mathrm{B}\mathcal{O}(X)$ is a simplicial copresheaf on $T$ 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 $[T, sSet]_{proj,prod}$. 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 $\infty$-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.
Fix a field $k$ of characteristic 0. We consider now the context of dg-geometry with its function algebras on ∞-stacks taking values in unbounded dg-algebras, exhibited by the adjoint (∞,1)-functors
For the discussion of Hochschild homology in this $\mathbf{H}$, 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 $k$ we have that
This is discussed in detail at model structure on dg-algebras in the sections Derived copowering and Derived powering.
By the above fact Pirashvili’s copowering definition of higher order Hochschild homology holds true in dg-geometry. For $X$ a manifold regarded as a topological space and then as a constant ∞-stack in $\mathbf{H}$ we have for any $A \in cdgAlg_k$
in $cdgAlg_k$.
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.
For $X$ a manifold, $\Omega^\bullet(X)$ its de Rham dg-algebra and $L X$ its free loop space, we have
We sketch the proof in terms of the above derived loop space technology.
Set $k = \mathbb{R}$. Write $LConst X \in \mathbf{H}$ for the constant ∞-stack on the homotopy type of $X$, regarded as a topological space $\simeq$ ∞Grpd. Then
is (…) the the $k$-valued singular cochain complex of $X$, which by the de Rham theorem is equivalent to the de Rham dg-algebra.
Since $LConst$ is a left exact (∞,1)-functor it commutes with forming free loop space objects and therefore
Since $LConst X$ is $\mathcal{O}$-perfect (…) we have by the above copowering-description of the Hochschild complexes that the cohomology of the loop space of $X$
is given by the Hochschild complex of the dg-algebra $\Omega^\bullet(X)$
Consider as before the categorical circle $S^1$ as the corresponding constant ∞-stack in $\mathbf{H}$. We describe the function $\infty$-algebra on $S^1$. 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
where on the right we have the ring of dual numbers over $k$, regarded as a dg-algebra with the odd generator in degree 1 and trivial differential.
Every ∞-groupoid is the (∞,1)-colimit over itself (as described there) of the (∞,1)-functor constant on the point. This (∞,1)-colimit is preserved by the left adjoint (∞,1)-functor $LConst : \infty Grpd \to \mathbf{H}$, so that we have
in $\mathcal{H}$. The (∞,1)-functor $\mathcal{O}$ is also left adjoint, so that
in $cdgAlg_k^\circ$. Since the point is representable, we have by the definition of $\mathcal{O}$ as the left $(\infty,1)$-Kan extension of the inclusion $(cdgAlg_k^-)^{op} \hookrightarrow (cdgAlg_k)^{op}$ that this is
This is the formula for the $(\infty,1)$-power of the cdg-algebra $k$ by by $\infty$-groupoid $S^1$. By the above fact, using that the circle is a finite $(\infty,1)$-groupoid, this is given by the cdg-algebra of polynomial differential forms on simplices of $S^1$
By a central theorem of rational homotopy theory (recalled at differential forms on simplices) this is equivalent to the singular cochains on the circle
But $S^1 \simeq \mathcal{B}\mathbb{Z}$ is a classifying space of a Lie algebra, so that this is a formal dg-algebra, equivalent to its cochain cohomology. Over the field $k$ of characteristic 0 this is
Therefore
This means that $Spec \mathcal{O}(S^1)$ is no longer the circle itself, but the odd line, regarded with its canonical $\mathbb{Z}$-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 $\mathcal{O}(X)$ encodes Kähler differential forms on $X$;
Hochschild cohomology of $\mathcal{O}(X)$ encodes multivector fields on $X$;
there are natural pairings between $HH_\bullet(\mathcal{O}(X), \mathcal{O}(X))$ and $HH^\bullet(\mathcal{O}(X), \mathcal{O}(X))$ that mimic the structure of the natural pairings between vector fields and differential forms on smooth manifold.
See (Tamarkin-Tsygan) and see at Kontsevich formality for more. This equivalence enters the construction of formal deformation quantization of Poisson manifolds.
One way to understand or interpret this conceptually is to regard the derived loop space object of a 0-truncated object $X$ to consist of infinitesimal loops in $X$.
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.
Let $A$ be an associative algebra over $k$. Recall the natural identification
of the first Hochschild homology of $A$ with coefficients in itself and degree-1 Kähler differential forms of $A$.
Write $\Omega^0(R/k) := R \simeq HH_0(R,R)$.
For $n \geq 2$ Write $\Omega^n(R/k) = \wedge^n_R \Omega(R/k)$ for the $n$-fold wedge product of $\Omega(R/k)$ with itself: the degree $n$-Kähler-differentials.
The isomorphism $\Omega^1(R/k) \simeq H_1(R,R)$ extends to a graded ring morphism
If the $k$-algebra $R$ is sufficiently well-behaved, then this morphism is an isomorphism that identifies the Hochschild homology of $R$ in degree $n$ with $\Omega^n(R/k)$ for all $n$:
(Hochschild-Kostant-Rosenberg theorem)
If $k$ is a field and $R$ a commutative $k$-algebra which is
essentially of finite type
smooth over $k$
then there is an isomorphism of graded $R$-algebras
Moreover, dually, there is an isomorphism of Hochschild cohomology with wedge products of derivations:
This is reviewed for instance as (Weibel, theorem 9.4.7) or as (Ginzburg, theorem 9.1.3).
The higher order Hochschild homology $\mathcal{O} (X^{S^d})$ of an object $X$ with respect to the $d$-sphere $S^d$ 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 $k = d+1$ .
For let $\Sigma^{d+1} = D^{d+1}\setminus \coprod_r D^{d+1}$ be the $(d+1)$-ball with $r$ small $d+1$-balls taken out. We have a cospan of boundary inclusions
in ∞Grpd and under $LConst : \infty Grpd \to \mathbf{H}$ then also in our (∞,1)-topos.
Applying the (∞,1)-topos internal hom $[-,X]$ or equivalent the (∞,1)-powering $X^{(-)}$ into a given object $X \in \mathbf{H}$ to this cospan produces the span
in $\mathbf{H}$. Then the integral transforms on sheaves
induced by these spans constitute the $E_n$-action on the function objects on $X^{S^d}$.
This was observed in (Ben-ZviFrancisNadler, corollary 6.8).
The next statement is known as the Deligne conjecture.
(Deligne conjecture)
Some historical comments on the Deligne conjecture.
Historically it was first found that there is the structure of a Gerstenhaber algebra on $HH^\bullet(A,A)$. 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 $HH^\bullet(A,A)$ lifts to an E2-algebra-structure on the cochain complex $C^\bullet(A,A)$.
In GerstenhaberVoronov (1994) a resolution of the Gerstenhaber algebra structure was given, but the relationship to $E_2$-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 $C^\bullet(A,A)$. This proved Deligne’s conjecture.
Various authors later further refined this result. A summary of this history can be found in (Hess).
In Hu-Kriz-Voronov (2003) it was further shown that for $A$ an En-algebra, $C^\bullet(A,A)$ is an $E_{n+1}$-algebra.
Notice that the identification of Hochschild (co)homology as coming from higher order free loop spaces makes all this structure manifest.
Let $T$ be the algebraic theory of ordinary associative algebras over a field $k$, regarded as an (∞,1)-algebraic theory and let $\mathbf{H}$ be the (∞,1)-topos of $(\infty,1)$-sheaves over a small site in $T Alg_\infty^{op}$.
Under the inverse image of the global section (∞,1)-geometric morphism and the homotopy hypothesis-equivalence
we may regard every topological space $X$ as a constant ∞-stack $LConst X$, an object in $\mathbf{H}$.
The function algebra on $LConst X$ is the cosimplicial algebra of singular cochains on $X$. Under the monoidal Dold-Kan correspondence it identifies with the cochain dg-algebra $C^\bullet(X)$ that computes the singular cohomology of $X$.
This has maybe been first made explicit by Bertrand Toën. Details are at function algebras on ∞-stacks.
The Hochschild homology of $C^\bullet(X)$ is the singular cohomology of the free loop space $L X$.
Apply the central identification $\mathcal{O} \mathcal{L}(LConst X) \simeq S^1 \cdot \mathcal{O}(LConst X)$. Then observe that the free loop space object $\mathcal{L} LConst X$ of the constant $\infty$-stack is the constant $\infty$-stack on the ordinary free loop space, because $LConst$ is a left exact (∞,1)-functor and because $\mathcal{L}X \simeq L X$ in Top. Then use by the above remark that $\mathcal{O} LConst L X$ is singular cochains on $L X$.
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 $X$ be a compact manifold oriented smooth manifold of dimension $d$. Write $C^\bullet(X)$ for the dg-algebra of cochains for singular cohomology of $X$. Write $L X$ for the topological free loop space of $X$ and $H_\bullet(L X)$ for its singular homology.
There is a linear isomorphism of degree $d$
This is due to (FelixThomasVigue-Poirrier, section 7)).
(Jones’ theorem)
There is an isomorphism
such that the canonical string topology BV-operator $\Delta$ of the BV-algebra $H_{\bullet + d}(L X)$ and the Connes coboundary? $B^\vee$ on $HH^{\bullet-d}(C^\bullet(X), C^\bullet(X)^{\vee})$ satisfy
This is due to (Jones).
The Connes coboundary? defines via the isomorphism $\mathbb{D}$ from above the structure of a BV-algebra on $HH^\bullet(C^\bullet(X), C^\bullet(X))$.
This is (Menichi, theorem 3).
There is an intrinsic circle action on Hochschild (co)chains. Passing to the cyclically invariant (co)chains yields cyclic (co)homology.
An exact sequence $0 \to N \to E \to R$ of $k$-modules where $E \to R$ is a surjective morphism of $k$-algebras is called a $k$-split extension or a Hochschild extension of $R$ by $E$ if the sequence is a split sequence as a sequence of $k$-modules.
Two extensions are equivalent if there is an isomorphism or $k$-algebra $E \stackrel{\simeq}{\to} E'$ that makes
commute.
Due to the $k$-splitness assumption there is an isomorphism of $k$-modules $E \simeq R \oplus N$ and this is equipped with a $k$-algebra structure such that the product on the $R$ direct summand is that of $R$. From this we find that the product on $E$ is of the form
where $f : R \otimes_k R \to N$ is some $k$-linear map. Since the product on $E$ is (by definition) associative, it follows that for $f$ that this satisfies for all $r_0, r_1, r_2 \in R$ the cocycle equation
as an equation in $N$. This says that $f$ must be a Hochschild cocycle
Conversely, every such cocycle yields a $k$-split extension of $R$ by $N$ this way:
For $R$ a $k$-algebra and $N$ an $R$-bimodule, equivalence classes of Hochschild extensions of $R$ by $N$ are in bijection with degree 2 Hochschild cohomology $HH^2(R,N)$.
See for instance Weibel, theorem 9.3.1.
As a special case of the above statement about extensions of $R$, we obtain a statement about deformation of $R$.
A standard problem is to deform a $k$-algebra $R$ by introducing a new “parameter” $t$ that squares to 0 – $t \cdot t = 0$ and a new product
From the above we see that this is the same as finding an $k$-split extension of $R$ by itself. So in particular such extensions are given by Hochschild cocycles $f \in HH^2(R,R)$.
See for instance Ginzburg, section 7 and for more see deformation quantization.
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 $(\infty,1)$-copowering operation manifest is in
The full $(\infty,1)$-categorical picture of Hochschild homology as the cohomology of derived free loop space objects is due to
based on
Specifically the dicussion of differential forms via such an $\infty$-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 $(HH^\bullet(A,A), HH_\bullet(A,A))$ 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.