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.

*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
[tex]V=\bigoplus_{i\in\mathbb Z} V_i[/tex]; and we'll call
[tex]V_p[/tex] the component of degree p. The category of graded vector
spaces come equipped with one rather natural endofunctor: the suspension
S. It works by [tex](SV)_p=V_{p+1}[/tex]. An *n*-ary operator of
degree *k* on a graded vector space *V* is a family of maps
[tex](V^{\otimes n})_{i-k}\to V_i[/tex].

*n*-ary higher homotopy operator will have degree 2-

*n*.

*n*-ary homotopy is a map

*n*. That means, that we can just as well view it as a map

*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 [tex](SV)^{\otimes n}[/tex] ends up. This is, in reality the degree

*n*slice of [tex]V^{\otimes n}[/tex], 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 [tex]m_i[/tex] is a
degree 1 map [tex](SV)^{\otimes n}\to SV[/tex]. 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?

*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 [tex]m_1[/tex], and the differential on [tex]V^{\otimes n}[/tex] is induced from this as the sum [tex]\sum_{i=1}^n 1^{\otimes i-1}\otimes m_1\otimes 1^{\otimes n-i}[/tex]. Now, the various versions of associating three elements under this multiplication we've defined are [tex]m_2(1\otimes m_2)[/tex] and [tex]m_2(m_2\otimes 1)[/tex]. Both of these are chain maps [tex]SV^{\otimes 3}\to SV[/tex] (chain maps since the Leibniz rule holds). So we'll want [tex]m_2(1\otimes m_2-m_2\otimes 1)=dm_3+m_3d[/tex], where the first

*d*is just [tex]m_1[/tex] and the second is [tex]m_1\otimes1\otimes1+1\otimes m_1\otimes1+1\otimes1\otimes m_2[/tex]. 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?

## Examples

First of all, any algebra is a differential graded algebra concentrated
in degree 0. So if [tex]V_i=0[/tex] for all non-zero *i*, and
[tex]m_2[/tex] 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 [tex]m_1,m_2[/tex] are the only
non-zero maps.

### Hochschild cohomology

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.

*M*over a ring

*R*, we set [tex]C_0=M[/tex] and [tex]C_i=M\otimes R^{\otimes i}[/tex].

Now, let's take some associative algebra *B*, and look at the graded
algebra [tex]A=B[\epsilon]/\epsilon^2[/tex] with
[tex]|\epsilon|=2-N[/tex]. 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 [tex]c\colon B^N\to B[/tex],
and define [tex]m_2[/tex] to be normal multiplication in A, and
[tex]m_i[/tex]=0 for all other i. Now, we can get a new A_{∞}
structure by setting [tex]\mu_i=m_i[/tex] for all [tex]i\neq N[/tex]
and [tex]\mu_N=m_N+\epsilon c[/tex].

_{∞}-conditions that

*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.

*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 [tex]N[/tex]. Then, the only relations that survive, since all but two operations vanish, are those where [tex]\mu_2[/tex] and [tex]\mu_N[/tex] are combined. This occurs precisely for the relations of degrees

*N+1*and

*N(N-1)*. For the latter case, we combine [tex]\epsilon c[/tex] with [tex]\epsilon c[/tex], which gives us something multiplied with [tex]\epsilon^2[/tex], thus vanishing. For the first relation, we receive instead

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.