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
[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].
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
[tex]d\colon V\to V[/tex] of degree 1, and a multiplication
[tex]\mu\colon V\otimes V\to V[/tex] of degree 0. The homotopy of
the associativity of the multiplication will be some 3-ary map h such
that [tex]\mu(1\otimes\mu+\mu\otimes1)=hd+dh[/tex], 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 [tex](V^{\otimes n})_k = (SV^{\otimes n})_{k-n}[/tex]
(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
[tex]V^{\otimes n}\to V[/tex] of degree 2-n. That means, that
we can just as well view it as a map
[tex](SV)^{\otimes n}\to SV[/tex] 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
[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?
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
[tex]m_1^2=0[/tex]. Next, we'll want it to fulfill the Leibniz rule,
so that it really does behave like a differential. So [tex]m_1\circ
m_2 = m_2(1\otimes m_1+m_1\otimes 1)[/tex]. 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
[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
[tex]\sum_{n=r+s+t}(-1)^{r+st}m_{r+t+1}\circ(1^{\otimes
r}\otimes m_s\otimes1^{\otimes t})=0[/tex]
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.
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 [tex]C_0=M[/tex] and
[tex]C_i=M\otimes R^{\otimes i}[/tex].
The face maps on this complex is defined as
[tex]\partial_0(m\otimes r_1\otimes\dots\otimes
r_n)=mr_1\otimes r_2\otimes\dots\otimes r_n[/tex]
[tex]\partial_i(m\otimes r_1\otimes\dots\otimes
r_n)=m\otimes\dots\otimes r_ir_{i+1}\otimes\dots\otimes
r_n[/tex]
[tex]\partial_n(m\otimes r_1\otimes\dots\otimes
r_n)=r_nm\otimes\dots\otimes r_{n-1}[/tex]
and the degeneracy maps just slot in a [tex]\otimes 1\otimes[/tex]
at the appropriate index.
We build a complex from this by just putting
[tex]d=\sum(-1)^i\partial_i[/tex]. The homology of the resulting
chain complex is called the Hochschild homology
[tex]HH_*(M,R)[/tex]. Dualizing everything, we get a cochain complex
of multilinear maps [tex]R^n\to M[/tex], and the face maps
[tex]\partial^0f(r_0,\dots,r_n)=r_0f(r_1,\dots,r_n)[/tex]
[tex]\partial^if(r_0,\dots,r_n)=f(r_0,\dots,r_ir_{i+1},\dots,r_n)[/tex]
[tex]\partial^nf(r_0,\dots,r_n)=f(r_0,\dots,r_{n-1})r_n[/tex]
and the degeneracy maps again just inserting a 1 at the appropriate
position. The homology of the chain complex we get from
[tex]d=\sum(-1)^i\partial^i[/tex] has homology the Hochschild
cohomology [tex]HH^*(R,M)[/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].
Suppose first that we picked our [tex]N=2[/tex]. Then we get, from the
A∞-conditions that
[tex]\mu_1^2=0[/tex] (sure, [tex]\mu_1=0[/tex] anyway...)
[tex]\mu_1\circ \mu_2=\mu_2(1\otimes \mu_1+\mu_1\otimes
1)[/tex] (no problem. [tex]\mu_1[/tex] is still 0)
[tex]\mu_2(\mu_2\otimes 1-1\otimes\mu_2)=0[/tex], which we
can expand to, and insert values to get
[tex](\mu_2(\mu_2\otimes-1\otimes\mu_2))(x,y,z)=[/tex]
[tex]x(yz)-(xy)z+\epsilon(c(x,y)z-xc(y,z))+\epsilon(c(xy,z)-c(x,yz))+\epsilon^2(\cdots)[/tex]
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
[tex]dc=0[/tex], 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
[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
[tex]m_2((-1)^N\epsilon c\otimes1-1\otimes\epsilon
c)+\sum_{i=1}^{N}\epsilon c(1^{\otimes i-1}\otimes m_2\otimes
1^{\otimes N-i})[/tex], or if we translate it to something readable,
involving an application to elements, it turns into
[tex]r_0c(r_1,\dots,r_N)+(-1)^Nc(r_1,\dots,r_{N-1})r_N+\sum_{i=1}^N(-1)^ic(r_0,\dots,r_ir_{i+1},\dots,r_N)[/tex]
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.