### Infinitely Categorified Calculus

#### Posted by John Baez

Vasiliy Dolgushev gave a nice talk here at UCR today. He started by explaining how the usual concepts of calculus must be *infinitely categorified* to apply to noncommutative geometry!

$\infty$-categories are pretty scary. But, they’re less scary when they’re **strict**: when all the laws holds as equations instead of ‘up to isomorphism’. And they’re even less scary when they’re also **linear**: when they’ve got a *vector space* of objects, a *vector space* of morphisms, a *vector space* of 2-morphisms and so on, with all the operations being *linear*. It turns out that a strict linear $\infty$-category is just a chain complex of vector spaces!

If you already know and love chain complexes, this revelation might leave you unmoved. But it really does explain a lot about what people do with chain complexes.

In particular, when people start taking ordinary concepts from linear algebra, replacing all the vector spaces in sight by chain complexes, and making all the usual axioms hold ‘up to coherent homotopy’, they’re really $\infty$-categorifying these concepts.

Here’s what happens to some familiar concepts when we do this:

- associative algebras become $A_\infty$-algebras
- Lie algebras become $L_\infty$-algebras
- commutative algebras become $E_\infty$-algebras

Now, calculus on a manifold is really just a heavy-duty version of linear algebra: we’ve got differential forms, and vector fields, and so on, and a bunch of linear operations. Recently the whole lot have been formalized into a single mathematical gadget called — boldly but aptly — a *‘calculus’*.

We can get a calculus not just from a smooth manifold but from any commutative algebra. When we use the algebra of smooth functions on a smooth manifold, we’re back where we started — but it’s really a vast generalization.

What if we start with a noncommutative algebra? Here it turns out we just get a ‘$Calc_\infty$ algebra’: in other words, an *infinitely categorified* calculus!

Let me start by saying what a ‘calculus’ is.

To motivate the definition, suppose $M$ is a smooth manifold. Let

$\Omega = \Gamma(\Lambda T^* M)$

be the vector space of **differential forms** on $M$ — the funny notation on the right-hand side mean ‘the space of smooth sections of the exterior algebra of the cotangent bundle of $M$’, which is nothing but the space of differential forms on $M$.

Similarly, let

$V = \Gamma(\Lambda TM)$

be the vector space of **multivector fields** on $M$. Multivector fields are a bit less famous than differential forms, but they’re also very important. We define them just like differential forms, but starting with the tangent bundle instead of the cotangent bundle.

Both $\Omega$ and $V$ have a lot of extra structure which we use all the time in differential geometry. This structure becomes the definition of a ‘calculus’.

What is this structure? For starters, $\Omega$ is a supercommutative differential graded algebra. This is a quick way of summarizing the familiar properties of the exterior derivative

$d: \Omega \to \Omega$

and wedge product

$\wedge : \Omega \otimes \Omega \to \Omega.$

What about $V$? This is also a supercommutative graded algebra, with its own wedge product:

$\wedge : V \otimes V \to V.$

But, there’s no ‘exterior derivative’ of multivector fields. As a consolation prize, there’s a bracket operation, the **Schouten bracket**:

$[\cdot, \cdot]: V \otimes V \to V.$

You can define this by saying it’s the unique linear map extending the usual Lie bracket of vector fields in a way that makes the graded algebra $V$ into a graded Poisson algebra — but be careful: since the bracket of vector fields (aka ‘1-vector fields’) is another vector field instead of a 2-vector field, this is a funny sort of graded Poisson algebra where the bracket has degree $-1$. This sort of thing is called a **Gerstenhaber algebra**.

So, we’ve got a supercommutative differential graded algebra $\Omega$ and a Gerstenhaber algebra $V$. But there’s more, because these two structures interact in a nice way!

For starters, the usual ‘Lie derivative’ operation where vector fields act on differential forms extends to an action

$L : V \otimes \Omega \to \Omega$

making $\Omega$ into a module of $V$, thought of as a graded Lie algebra with bracket of degree $-1$. The usual ‘interior product’ operation of vector fields on differential forms also extends to an operation

$i: V \otimes \Omega \to \Omega .$

And, a bunch of identities hold for all multivector fields $v,w \in V$:

$[L_v, L_w] = L_{[v,w]}$

$[i_v, L_w] = i_{[v,w]}$

$L_{v \wedge w} = i_v L_w + (-1)^{deg v} i_w L_v$

$[L_v, d] = 0$

$i_v i_w = i_{v \wedge w}$

The brackets here, except for the bracket in $V$, are supercommutators.

Taking this all together, we get the definition of a **calculus**: a supercommutative differential graded algebra $\Omega$ and a Gerstenhaber algebra $V$ together with actions $L$, $i$ where $L$ makes $\Omega$ into a module of $V$ and the above identities hold.

(I hope I have everything here! What happened to the Weil identity $L_v = [d, i_v]$, familiar for vector fields $v$?)

Now the fun starts…

For any associative algebra $A$ we can define a chain complex of **Hochschild chains**:

$C_{\bullet}(A,A) = A \otimes A^{\otimes \bullet}$

and a cochain complex of **Hochschild cochains**:

$C^{\bullet}(A,A) = hom(A^{\otimes \bullet}, A)$

Taking homology and cohomology, respectively, we get **Hochschild homology**:

$H_{\bullet}(A,A)$

and **Hochschild cohomology**:

$H^{\bullet}(A,A)$

It turns out that if $A$ is the algebra of smooth functions on a compact manifold, the Hochschild homology gives the differential forms on $M$:

$H_{\bullet}(A,A) \cong \Omega$

while the Hochschild cohomology gives the multivector fields:

$H^{\bullet}(A,A) \cong V .$

So, in this case $H_{\bullet}(A,A)$ and $H^{\bullet}(A,A)$ form a calculus. But it’s always better to do things at the level of co/chains instead of just at the level of co/homology, whenever possible. So, this raises the question of whether $C_{\bullet}(A,A)$ and $C^{\bullet}(A,A)$ form a calculus as well, in a way that reduces to our familiar calculus on $\Omega$ and $V$ when we take co/homology.

And, since Hochschild co/chains make sense for any *associative* algebra, not just any *commutative* one, we should wonder if we also get a calculus in the noncommutative case!

The answer, in short, is *yes, if we are willing to infinitely categorify the concept of calculus!*

In other words, there’s a concept of $Calc_\infty$-algebra, which is like a calculus but where all the usual laws hold up to chain homotopy — and these chain homotopies themselves satisfy all the right laws up to chain homotopy, ad infinitum. And, we have:

Theorem.For any associative algebra $A$, the Hochschild chains $C_{\bullet}(A,A)$ and Hochschild cochains $C^{\bullet}(A,A)$ naturally form a $Calc_\infty$-algebra. This in turn makes $H_{\bullet}(A,A)$ and $H^{\bullet}(A,A)$ into a calculus, in such a way that when $A$ is the algebra of smooth functions on a compact manifold, we recover the usual calculus of differential forms and multivector fields.

A lot of work by a lot of people went into proving this result. For example, the differential

$d: C_p(A,A) \to C_{p+1}(A,A)$

is the ‘$B$’ operator invented by Connes in his work on noncommutative geometry. Tamarkin later showed that $C^{\bullet}(A,A)$ is a $G_\infty$-algebra, which is the infinitely categorified version of a Gerstenhaber algebra. (In fact, in this case all the differential graded Lie algebra axioms hold ‘on the nose’, as equations — it’s the other rules that hold only up to coherent homotopy.) The proof of this was deep: it used the Deligne’s conjecture (first proved by Kontsevich) and the formality of the little discs operad (first proved by Tarmakin himself). For more see this:

- Vasiliy Dolgushev, Dmitry Tamarkin, and Boris Tsygan, The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal.

But, apparently this paper pulled it all together:

- Maxim Kontsevich and Yan Soilbelman, Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I

By the way — in case you’re curious, the boundary map in Hochschild homology is given by:

$d(b \otimes a_1 \otimes \cdots \otimes a_n) =$ $b a_1 \otimes a_2 \cdots \otimes a_n - b \otimes a_1 a_2 \otimes \cdots \otimes a_n + \cdots + (-1)^{n-1} b \otimes a_1 \otimes \cdots a_{n-1} a_n +$ $(-1)^n a_n b \otimes a_1 \otimes \cdots \otimes a_{n-1}$

while the coboundary map in Hochschild cohomology is given by:

$(\delta f)(a_1, \dots, a_n) =$ $a_1 f(a_2, \dots, a_n) - f(a_1 a_2, \dots, a_n) + \cdots + (-1)^{n-1} f(a_1, \dots, a_{n-1} a_n) +$ $(-1)^n f(a_1, \dots, a_{n-1}) a_n .$

I should also correct a slight oversimplification above. When $A$ is the algebra of algebraic functions on a smooth affine variety, $H_\bullet(A,A)$ is isomorphic to the space of differential forms on $M$, while $H^\bullet(A,A)$ is isomorphic to the space of multivector fields. A similar thing holds when $A$ is the algebra of smooth functions on a compact manifold, but apparently some analysis intervenes, and we need to use ‘continuous’ Hochschild co/homology to make these statements hold.

## Re: Infinitely Categorified Calculus

But

Visnota differential graded Lie algebra!If the proof uses a conjecture, does this mean that it hasn’t actually been proved? Or is this the proved one? (Wikipedia says that there are about 7 Deligne’s Conjectures, one of which has been proved.)