This blogpost is a running log of my thoughts while reading a couple of papers by Bernhard Keller. I recommend anyone reading this and feeling interest to hit the arXiv and search for his introductions to A∞-algebras. Especially math.RA/9910179 serves as a basis for this post.
If you do enough of a particular brand of homotopy theory, you’ll sooner or later encounter algebras that occur somewhat naturally, but which aren’t necessarily associative as such, but rather only associative up to homotopy. The first obvious example is that of a loop space, viewed as a group: associativity only holds after you impose equivalence between homotopic loops.
In fact, if we start with a based space — the normal situation in original homotopy of topological spaces — and start looking at the space of all loops based in the basepoint of the space. A loop is simply an image of the circle in the topological space, with the circle based in 0 and running up to 2π. Then we know we can compose loops, by just running through first one on half the time, and then the other on the second half the time. So from 0 to π we run through one loop, and then from π to 2π we run through the other.
However, we now get genuinely different loops — seen as functions — from the different ways to compose three loops. Let the loops be called f,g,h. Then f(gh) is a different loop from (fg)h, since in the first loop, f happens from 0 to π and in the second, f happens only from 0 to π/2. But we were doing homotopy! So we have some continuous map, named the homotopy, that goes from f(gh) to (fg)h. So then everything is alright after all. We don’t have strict associativity in the loop space, but we have associativity up to homotopy.
But now, if we look at the different ways to compose four loops, we get odd things happening. We’d have f(g(hi)) and ((fg)h)i as to extremal versions, and then two different ways to move between them. One way is to go
f(g(hi)) → (fg)(hi) → ((fg)h)i and the other one is to go
f(g(hi)) → f((gh)i) → (f(gh))i → ((fg)h)i
So we’re stuck in the same place again, but with a higher level set of problems. Not really a big problem: we’ll simply require there to be a homotopy between these two paths. So we slot in a continuous function that takes care of that.
But then, if we look at the ways to compose five loops, we end up getting homotopies that form a spherical shell, and the same problem that we can do things in different way. But we can always make sure these are homotopical as well. And so on. The different homotopies needed are called associahedra and were introduced by Stasheff. A topological space that admits all these is called an A∞-space. Moreover, if a topological space admits an A∞-structure, then it is homotopy equivalent to a loop space of something.
But hang on a second. I’m doing algebra, not topology. What good is this?
Well, we certainly have a concept of associativity for algebras. And we do, once we start juggling the right kind of objects, have a concept of homotopies in an algebraic setting. So, we’ll try to mimic all of this but with a suitable setting to be able to talk about algebras that are simultaneously chain complexes.
A-infinity algebras enter the stage
So, we’ll want an algebra over some field k. We’ll start of gently, introducing it first as a graded vectorspace ; and we’ll call the component of degree p. The category of graded vector spaces come equipped with one rather natural endofunctor: the suspension S. It works by . An n-ary operator of degree k on a graded vector space V is a family of maps .
We want to tune a family of maps corresponding to the composition and higher homotopies in the topological situation, as well as handling the differential; which we want there just because we’re doing homological algebra and see a graded space. So we’ll want a differential
of degree 1, and a multiplication of degree 0. The homotopy of the associativity of the multiplication will be some 3-ary map h such that , whereby the lefthand side has degree 0, and the righthand side has degree 1 more than the degree of h, since the differential has degree 1. Thus, for each of the higher homotopies, we’ll fall one degree step. All in all, the n-ary higher homotopy operator will have degree 2-n.
We note that (just check the degrees on both sides…), and so that this, maybe slightly artificial looking, degree condition on the higher homotopies can be handled rather neatly. Each n-ary homotopy is a map
of degree 2-n. That means, that we can just as well view it as a map
of some degree; since the S operator only changes the degrees within V. We can find the degree of this map by looking at where the degree 0 slice of ends up. This is, in reality the degree n slice of , and thus ends up in degree 2 of V, which is to say that it ends up in degree 1 of SV. So the degree 2-n map defined on V turns into a degree 1 map of SV; which is rather neat.
So, each interesting map — let’s call them all is a degree 1 map . This family of maps is a part of the structure definition of an A∞-algebra. The rest of its definition is the set of properties we want these maps to fulfill. And what would those be?
First of all, we want it to be a chain complex when we’re done, so that we can use it to do homological stuff. So we’ll want . Next, we’ll want it to fulfill the Leibniz rule, so that it really does behave like a differential. So . And we want the associativity to hold – but only up to homotopy. That f and g are homotopic means that there is some chain map h such that f-g=hd+dh for the differentials on the chain complexes that f and g go between. Now, the differential on V we take to be our , and the differential on is induced from this as the sum . Now, the various versions of associating three elements under this multiplication we’ve defined are and . Both of these are chain maps (chain maps since the Leibniz rule holds). So we’ll want , where the first d is just and the second is . If we go on with all the higher associahedra, and choose our signs in a neat way, we’ll end up with the generic condition that
So that gives us some sort of abstract feel for what an A∞-algebra is. Do we know any examples? Can we construct any?
First of all, any algebra is a differential graded algebra concentrated in degree 0. So if for all non-zero i, and is the only non-zero structure map, then we recover the normal associative algebras.
A differential graded algebra is an A∞-algebra, associative and not only up to homotopy. This is the same as all higher homotopies vanishing, so it is the case where are the only non-zero maps.
Inspired by the success of simplicial complexes in algebraic topology, the inspiration pops up to try and introduce similar things in other theories. Thus, the idea of simplicial objects appears – which are defined as a functor from the category of finite ordered sets of integers with nondecreasing monotone functions as the arrows. In this setting, face maps and degeneracy maps get introduced: the face maps miss precisely one element, and the degeneracy maps duplicate precisely one element. This corresponds closely to our intuition of faces of simplices and degenerate simplices. With a simplicial objects theory, the differential in a chain complex ends up being just the sum of faces with appropriate signs. The whole theory varies far more than here indicated though.
Suppose now we want to construct a cohomology theory which includes this at its core. One method of arriving there is the theory of Hochschild cohomology. This is built basically the same way as any other cohomology theory, with the salient difference that the differentials and degeneracy maps are chosen differently. So to a bimodule M over a ring R, we set and .
The face maps on this complex is defined as
and the degeneracy maps just slot in a at the appropriate index.
We build a complex from this by just putting . The homology of the resulting chain complex is called the Hochschild homology . Dualizing everything, we get a cochain complex of multilinear maps , and the face maps
and the degeneracy maps again just inserting a 1 at the appropriate position. The homology of the chain complex we get from has homology the Hochschild cohomology .
Now, let’s take some associative algebra B, and look at the graded algebra with . Note to the attentive reader. This construction is very reminiscent of a graded version of what sigfpe is doing with practical synthetic differential geometry. We furthermore pick some multilinear map , and define to be normal multiplication in A, and =0 for all other i. Now, we can get a new A∞ structure by setting for all and .
Suppose first that we picked our . Then we get, from the A∞-conditions that
(no problem. is still 0)
, which we can expand to, and insert values to get
where x(yz)-(xy)z=0 since B is associative anyway, and by multilinearity, we note that c(x,y)z=c(x,yz) and xc(y,z)=c(xy,z), so everything cancels out. Thus this is fulfilled by the preconditions and does not bring any additional information.
Being a Hochschild 2-cocycle, however, only mandates that , which using the definitions means that dc(x,y,z)=xc(y,z)-c(xy,z)+c(x,yz)-c(x,y)z=0, which fits perfectly with our observation. Suppose now that we pick some other . Then, the only relations that survive, since all but two operations vanish, are those where and are combined. This occurs precisely for the relations of degrees N+1 and N(N-1). For the latter case, we combine with , which gives us something multiplied with , thus vanishing. For the first relation, we receive instead
, or if we translate it to something readable, involving an application to elements, it turns into
which looks very much like the Hochschild differential on cochains.
So, this is an A∞-algebra precisely when the chosen map c is a Hochschild cocycle.
If you’ve read this far, I’m impressed. If you’ve understood what I’ve been saying, I’m even more impressed – and probably will run across you sooner or later at some conference or other.