- #1
John Baez
Also available as http://math.ucr.edu/home/baez/week258
November 25, 2007
This Week's Finds in Mathematical Physics (Week 258)
John Baez
Happy Thanksgiving! Today I'll talk about a conjecture by Deligne
on Hochschild cohomology and the little 2-cubes operad.
But first I'll talk about... dust!
I began "week257" with some chat about about dust in a binary star
system called the Red Rectangle. So, it was a happy coincidence
when shortly thereafter, I met an expert on interstellar dust.
I was giving some talks at James Madison University in Harrisonburg,
Virginia. They have a lively undergraduate physics and astronomy
program, and I got a nice tour of some labs - like Brian Utter's
granular physics lab.
It turns out nobody knows the equations that describe the flow of
grainy materials, like sand flowing through an hourglass. It's a
poorly understood state of matter! Luckily, this is a subject where
experiments don't cost a million bucks.
Brian Utter has a nice apparatus consisting of two clear plastic
sheets with a bunch of clear plastic disks between them - big
"grains". And, he can make these grains "flow". Since they're
made of a material that changes its optical properties under stress,
you can see "force chains" flicker in and out of existence as lines
of grains get momentarily stuck and then come unstuck!
These force chains look like bolts of lightning:
1) Brian Utter and R. P. Behringer, Self-diffusion in dense
granular shear flows, Physical Review E 69, 031308 (2004).
Also available as arXiv:cond-mat/0402669.
I wonder if conformal field theory could help us understand these
simplified 2-dimensional models of granular flow, at least near some
critical point between "stuck" and "unstuck" flow. Conformal field
theory tends to be good at studying critical points in 2d physics.
Anyway, I'm digressing. After looking at a chaotic double pendulum
in another lab, I talked to Harold Butner about his work using radio
astronomy to study interstellar dust.
He told me that the dust in the Red Rectangle contains a lot of PAHs -
"polycyclic aromatic hydrocarbons". These are compounds made of
hexagonal rings of carbon atoms, with some hydrogens along the edges.
On Earth you can find PAHs in soot, or the tarry stuff that forms
in a barbecue grill. Wherever carbon-containing materials suffer
incomplete combustion, you'll find PAHs.
Benzene has a single hexagonal ring, with 6 carbons and 6 hydrogens -
a wonder of quantum resonance. You've probably heard about
naphthalene, which is used for mothballs. This consists of two
hexagonal rings stuck together. True PAHs have more. "Anthracene"
and "phenanthrene" consist of three rings. "Napthacene", "pyrene",
"triphenylene" and "chrysene" consist of four, and so on:
2) Wikipedia, Polycyclic aromatic hydrocarbon,
http://en.wikipedia.org/wiki/Polycyclic_aromatic_hydrocarbon
In 2004, a team of scientists discovered anthracene and pyrene in the
Red Rectangle! This was first time such complex molecules had been
found in space:
3) Uma P. Vijh, Adolf N. Witt, and Karl D. Gordon, Small polycyclic
aromatic hydrocarbons in the Red Rectangle, The Astrophysical
Journal, 619 (2005) 368-378.
By now, lots of organic molecules have been found in interstellar
or circumstellar space. There's a whole "ecology" of organic
chemicals out there, engaged in complex reactions. Life on planets
might someday be seen as just an aspect of this larger ecology.
I've read that about 10% of the interstellar carbon is in the form
of PAHs - big ones, with about 50 carbons per molecule. They're
common because they're incredibly stable. They've even been found
riding the shock wave of a supernova explosion!
PAHs are also found in meteorites called "carbonaceous chondrites".
These space rocks contain just a little carbon - about 3% by weight.
But, 80% of this carbon is in the form of PAHs.
Here's an interview with a scientist who thinks PAHs were important
precursors of life on Earth:
5) Aromatic world, interview with Pascale Ehrenfreund,
Astrobiology Magazine, available at
http://www.astrobio.net/news/modules.php?op=modload&name=News&file=article&sid=1992
And here's a book she wrote, with a chapter on organic molecules
in space:
6) Pascale Ehrenfreud, editor, Astrobiology: Future Perspectives,
Springer Verlag, 2004.
Harold Butner also told me about dust disks that have been seen around
the nearby stars Vega and Epsilon Eridani. By examining these disks,
we may learn about planets and comets orbiting these stars. Comets
emit a lot of dust, and planets affect its motion.
Mathematicians will be happy to know that *symplectic geometry*
is required to simulate the motion of this dust:
7) A. T. Deller and S. T. Maddison, Numerical modelling of
dusty debris disks, Astrophys. J. 625 (2005), 398-413.
Also available as arXiv:astro-ph/0502135
Okay... now for a bit about Hochschild cohomology. I want to
outline a conceptual proof of Deligne's conjecture that the
Hochschild cochain complex is an algebra for the little 2-cubes
operad. There are a bunch of proofs of this by now. Here's a
great introduction to the story:
8) Maxim Kontsevich, Operads and motives in deformation
quantization, available as arXiv:math/9904055.
I was inspired to seek a more conceptual proof by some conversations
I had with Simon Willerton in Sheffield this summer, and this paper
of his:
9) Andrei Calderou and Simon Willerton, The Mukai pairing, I:
a categorical approach, available as arXiv:0707.2052
But, while trying to write up a sketch of this more conceptual
proof, I discovered that it had already been worked out:
10) P. Hu, H. Kriz and A. A. Voronov, On Kontsevich's Hochschild
cohomology conjecture, available at arXiv:math.AT/0309369.
This was a bit of a disappointment - but also a relief. It
means I don't need to worry about the technical details: you
can just look them up! Instead, I can focus on sketching the
picture I had in mind.
If you don't know anything about Hochschild cohomology, don't worry!
It only comes in at the very end. In fact, the conjecture
follows from something simpler and more general. So, what you
really need is a high tolerance for category theory, homological
algebra and operads.
First, suppose we have any monoidal category. Such a category
has a tensor product and a unit object, which we'll call I. Let
end(I) be the set of all endomorphisms of this unit object.
Given two such endomorphisms, say
f: I -> I
and
g: I -> I
we can compose them, getting
f o g: I -> I
This makes end(I) into a monoid. But we can also tensor f and
g, and since I tensor I is isomorphic to I in a specified way,
we can write the result simply as
f tensor g: I -> I
This makes end(I) into a monoid in another, seemingly different
way.
Luckily, there's a thing called the Eckmann-Hilton argument which
says these two ways are equal. It also says that end(I) is a
*commutative* monoid! It's easiest to understand this argument
if we write f o g vertically, like this:
f
g
and f tensor g horizontally, like this:
f g
Then the Eckmann-Hilton argument goes as follows:
f 1 f g 1 g
= = g f = =
g g 1 1 f f
Here 1 means the identity morphism 1: I -> I. Each step in the
argument follows from standard stuff about monoidal categories.
In particular, an expression like
f g
h k
is well-defined, thanks to the interchange law
(f tensor g) o (h tensor k) = (f o h) tensor (g o k)
If we want to show off, we can say the interchange law says we've got
a "monoid in the category of monoids" - and the Eckmann-Hilton
argument shows this is just a monoid. See "week100" for more.
But the cool part about the Eckmann-Hilton argument is that we're
just moving f and g around each other. So, this argument has a
topological flavor! Indeed, it was first presented as an argument
for why the second homotopy group is commutative. It's all about
sliding around little rectangles... or as we'll soon call
them, "little 2-cubes".
Next, let's consider a version of this argument that holds only
"up to homotopy". This will apply when we have not a *set*
of morphisms from any object X to any object Y, but a *chain
complex* of morphisms.
Instead of getting a set end(I) that's a commutative monoid, we'll
get a cochain complex END(I) that's a commutative monoid "up to
coherent homotopy". This means that the associative and commutative
laws hold up to chain homotopies, which satisfy their own laws up to
homotopy, ad infinitum.
More precisely, END(I) will be an "algebra of the little 2-cubes
operad". This implies that for every configuration of n little
rectangles in a square:
---------------------
| |
| ----- |
| ----- | | |
|| | | | |
|| | | | |
| ----- | | |
| ----- |
| ---------------- |
| | | |
| ---------------- |
| |
---------------------
we get an n-ary operation on END(I). For every homotopy between
such configurations:
--------------------- ---------------------
| | | ----- |
| ----- | || | ---- |
| ----- | | | || | | | |
|| | | | | || | | | |
|| | | | | || | | | |
| ----- | | | ---> | ----- | | |
| ----- | | ----- |
| ---------------- | | ------- |
| | | | | | | |
| ---------------- | | ------- |
| | | |
--------------------- ---------------------
we get a chain homotopy between n-ary operations on END(I). And
so on, ad infinitum.
For more on the little 2-cubes operad, see "week220". In fact,
what I'm trying to do now is understand some mysteries I described
in that article: weird relationships between the little 2-cubes
operad and Poisson algebras.
But never mind that stuff now. For now, let's see how easy it is to
find situations where there's a chain complex of morphisms between
objects. It happens throughout homological algebra!
If that sounds scary, you should refer to a book like this as you
read on:
10) Charles Weibel, An Introduction to Homological Algebra,
Cambridge U. Press, Cambridge, 1994.
Okay. First, suppose we have an abelian category. This provides a
context in which we can reason about chain complexes and cochain
complexes of objects. A great example is the category of R-modules
for some ring R.
Next, suppose every object X in our abelian category has an
"projective resolution" - that is, a chain complex
d_0 d_1 d_2
X_0 <--- X_1 <--- X_2 <--- ...
where each guy X_i is projective, and the homology groups
ker (d_i)
H_i = -------------
im (d_{i-1})
are zero except for H_0, which equals X. You should think of
a projective resolution as a "puffed-up" version of X that's
better for mapping out of than X itself.
Given this, besides the usual set hom(X,Y) of morphisms from the
object X to the object Y, we also get a cochain complex which I'll
call the "puffed-up hom":
HOM(X,Y)
How does this work? Simple: replace X by a chosen projective
resolution
X^0 <--- X^1 <--- X^2 <--- ...
and then map this whole thing to Y, getting a cochain complex
hom(X^0,Y) ---> hom(X^1,Y) ---> hom(X^2,Y) ---> ...
This chain complex is the puffed-up hom, HOM(X,Y).
Now, you might hope that the puffed-up hom gives us a new category
where the hom-sets are actually cochain complexes. This is morally
true, but the composition
o: HOM(X,Y) x HOM(Y,Z) -> HOM(X,Z)
probably isn't associative "on the nose". However, I think it should
be associative up to homotopy! This homotopy probably won't satisfy
the law you'd hope for - the pentagon identity. But, it should
satisfy the pentagon identity up to homotopy! In fact, this should
go on forever, which is what we mean by "up to coherent homotopy".
This kind of situation is described by an infinite sequence of shapes
called "associahedra" discovered by Stasheff (see "week144").
If this is the case, instead of a category we get an "A-infinity
category": a gadget where the hom-sets are cochain complexes and the
associative law holds up to coherent homotopy. I'm not sure the
puffed-up hom gives an A-infinity category, but let's assume so and
march on.
Suppose we take any object X in our abelian category. Then we get
a cochain complex
END(X) = HOM(X,X)
equipped with a product that's associative up to coherent homotopy.
Such a thing is known as an "A-infinity algebra". It's just an
A-infinity category with a single object, namely X.
Next suppose our abelian category is monoidal. (To get the tensor
product to play nice with the hom, assume tensoring with any object
is right exact.) Let's see what happens to the Eckmann-Hilton
argument. We should get a version that holds "up to coherent
homotopy".
Let I be the unit object, as before. In addition to composition:
o: END(I) x END(I) -> END(I)
tensoring should give us another product:
tensor: END(I) x END(I) -> END(I)
which is also associative up to coherent homotopy. So, END(I) should
be an A-infinity algebra in two ways. But, since composition
and tensoring in our original category get along nicely:
(f tensor g) o (h tensor k) = (f o h) tensor (g o k)
END(I) should really be an A-infinity algebra in the category of
A-infinity algebras!
Given this, we're almost done. A monoid in the category of monoids
is a commutative monoid - that's another way of stating what the
Eckmann-Hilton argument proves. Similarly, an A-infinity algebra in
the category of A-infinity algebras is an algebra of the little
2-cubes operad. So, END(I) is an algebra of the little 2-cubes
operad.
Now look at an example. Fix some algebra A, and take our
monoidal abelian category to have:
A-A bimodules as objects
A-A bimodule homomorphisms as morphisms
Here the tensor product is the usual tensor product of bimodules,
and the unit object I is A itself. And, as Simon Willerton pointed
out to me, END(I) is a chain complex whose homology is familiar:
it's the "Hochschild homology" of A.
So, the cochain complex for Hochschild cohomology is an algebra of
the little 2-cubes operad! But, we've seen this as a consequence
of a much more general fact.
To wrap up, here are a few of the many technical details I glossed
over above.
First, I said a projective resolution of X is a puffed-up version of
X that's better for mapping out of. This idea is made precise
in the theory of model categories. But, instead of calling it a
"puffed-up version" of X, they call it a "cofibrant replacement" for
X. Similarly, a puffed-up version of X that's better for mapping
into is called a "fibrant replacement".
For a good introduction to this, try:
11) Mark Hovey, Model Categories, American Mathematical Society,
Providence, Rhode Island, 1999.
Second, I guessed that for any abelian category where every object has
a projective resolution, we can create an A-infinity category using
the puffed-up hom, HOM(X,Y). Alas, I'm not really sure this is true.
Hu, Kriz and Voronov consider a more general situation, but what I'm
calling the "puffed-up hom" should be a special case of their "derived
function complex". However, they don't seem to say what weakened sort
of category you get using this derived function complex - maybe an
A-infinity category, or something equivalent like a quasicategory or
Segal category? They somehow sidestep this issue, but to me it's
interesting in its own right.
At this point I should mention something well-known that's similar
to what I've been talking about. I've been talking about the
"puffed-up hom" for an abelian category with enough projectives.
But most people talk about "Ext", which is the cohomology of the
puffed-up hom:
Ext^i(X,Y) = H^i(HOM(X,Y))
And, while I want
END(X) = HOM(X,X)
to be an A-infinity algebra, most people seem happy to have
Ext(X) = H(HOM(X,X))
be an A-infinity algebra. Here's a reference:
12) D.-M. Lu, J. H. Palmieri, Q.-S. Wu and J. J. Zhang,
A-infinity structure on Ext-algebras, available as
arXiv:math.KT/0606144.
I hope they're secretly getting this A-infinity structure on
H(HOM(X,X)) from an A-infinity structure on HOM(X,X). They don't
come out and say this is what they're doing, but one promising
sign is that they use a theorem of Kadeishvili, which says that
the cohomology of an A-infinity algebra is an A-infinity algebra.
Finally, the really interesting part: how do we make an A-infinity
algebra in the category of A-infinity algebras into an algebra of
the little 2-cubes operad? This is the heart of the "homotopy
Eckmann-Hilton argument".
I explained operads, and especially the little k-cubes operad,
back in "week220". The little k-cubes operad is an operad in
the world of topological spaces. It has one abstract n-ary operation
for each way of sticking n little k-dimensional cubes in a big
one, like this:
---------------------
| |
| ----- |
| ----- | | |
|| | | | |
|| | | | | typical
| ----- | | | 3-ary operation in the
| ----- | little 2-cubes operad
| ---------------- |
| | | |
| ---------------- |
| |
---------------------
A space is called an "algebra" of this operad if these abstract
n-ary operations are realized as actual n-ary operations on the
space in a consistent way. But, when we study the homology
of topological spaces, we learn that any space gives a chain complex.
This let's us convert any operad in the world of topological spaces
into an operad in the world of chain complexes. Using this, it also
makes sense to speak of a *chain complex* being an algebra of the
little k-cubes operad. For that matter, a cochain complex.
Let's use "E(k)" to mean the chain complex version of the little
k-cubes operad.
An "A-infinity algebra" is an algebra of a certain operad called
A-infinity. This isn't quite the same as the operad E(1), but it's
so close that we can safely ignore the difference here: it's
"weakly equivalent".
Say we have an A-infinity algebra in the category of A-infinity
algebras. How do we get an algebra of the little 2-cubes operad,
E(2)?
Well, there's a way to tensor operads, such that an algebra of
P tensor Q is the same as a P-algebra in the category of Q-algebras.
So, an A-infinity algebra in the category of A-infinity algebras is
the same as an algebra of
A-infinity tensor A-infinity
Since A-infinity and E(1) are weakly equivalent, we can turn this
algebra into an algebra of
E(1) tensor E(1)
But there's also an obvious operad map
E(1) tensor E(1) -> E(2)
since the product of two little 1-cubes is a little 2-cube.
This too is a weak equivalence, so we can turn our algebra of
E(1) tensor E(1) into an algebra of E(2).
The hard part in all this is showing that the operad map
E(1) tensor E(1) -> E(2)
is a weak equivalence. In fact, quite generally, the map
E(k) tensor E(k') -> E(k+k')
is a weak equivalence. This is Proposition 2 in the paper by
Hu, Kriz and Voronov, based on an argument by Gerald Dunn:
13) Gerald Dunn, Tensor products of operads and iterated loop
spaces, Jour. Pure Appl. Alg 50 (1988), 237-258.
Using this, they do much more than what I've sketched: they
prove a conjecture of Kontsevich which says that the Hochschild
complex of an algebra of the little k-cubes operad is an algebra
of the little (k+1)-cubes operad!
That's all for now. Sometime I should tell you how this is related
to Poisson algebras, 2d TQFTs, and much more. But for now, you'll
have to read that in Kontsevich's very nice paper.
-----------------------------------------------------------------------
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November 25, 2007
This Week's Finds in Mathematical Physics (Week 258)
John Baez
Happy Thanksgiving! Today I'll talk about a conjecture by Deligne
on Hochschild cohomology and the little 2-cubes operad.
But first I'll talk about... dust!
I began "week257" with some chat about about dust in a binary star
system called the Red Rectangle. So, it was a happy coincidence
when shortly thereafter, I met an expert on interstellar dust.
I was giving some talks at James Madison University in Harrisonburg,
Virginia. They have a lively undergraduate physics and astronomy
program, and I got a nice tour of some labs - like Brian Utter's
granular physics lab.
It turns out nobody knows the equations that describe the flow of
grainy materials, like sand flowing through an hourglass. It's a
poorly understood state of matter! Luckily, this is a subject where
experiments don't cost a million bucks.
Brian Utter has a nice apparatus consisting of two clear plastic
sheets with a bunch of clear plastic disks between them - big
"grains". And, he can make these grains "flow". Since they're
made of a material that changes its optical properties under stress,
you can see "force chains" flicker in and out of existence as lines
of grains get momentarily stuck and then come unstuck!
These force chains look like bolts of lightning:
1) Brian Utter and R. P. Behringer, Self-diffusion in dense
granular shear flows, Physical Review E 69, 031308 (2004).
Also available as arXiv:cond-mat/0402669.
I wonder if conformal field theory could help us understand these
simplified 2-dimensional models of granular flow, at least near some
critical point between "stuck" and "unstuck" flow. Conformal field
theory tends to be good at studying critical points in 2d physics.
Anyway, I'm digressing. After looking at a chaotic double pendulum
in another lab, I talked to Harold Butner about his work using radio
astronomy to study interstellar dust.
He told me that the dust in the Red Rectangle contains a lot of PAHs -
"polycyclic aromatic hydrocarbons". These are compounds made of
hexagonal rings of carbon atoms, with some hydrogens along the edges.
On Earth you can find PAHs in soot, or the tarry stuff that forms
in a barbecue grill. Wherever carbon-containing materials suffer
incomplete combustion, you'll find PAHs.
Benzene has a single hexagonal ring, with 6 carbons and 6 hydrogens -
a wonder of quantum resonance. You've probably heard about
naphthalene, which is used for mothballs. This consists of two
hexagonal rings stuck together. True PAHs have more. "Anthracene"
and "phenanthrene" consist of three rings. "Napthacene", "pyrene",
"triphenylene" and "chrysene" consist of four, and so on:
2) Wikipedia, Polycyclic aromatic hydrocarbon,
http://en.wikipedia.org/wiki/Polycyclic_aromatic_hydrocarbon
In 2004, a team of scientists discovered anthracene and pyrene in the
Red Rectangle! This was first time such complex molecules had been
found in space:
3) Uma P. Vijh, Adolf N. Witt, and Karl D. Gordon, Small polycyclic
aromatic hydrocarbons in the Red Rectangle, The Astrophysical
Journal, 619 (2005) 368-378.
By now, lots of organic molecules have been found in interstellar
or circumstellar space. There's a whole "ecology" of organic
chemicals out there, engaged in complex reactions. Life on planets
might someday be seen as just an aspect of this larger ecology.
I've read that about 10% of the interstellar carbon is in the form
of PAHs - big ones, with about 50 carbons per molecule. They're
common because they're incredibly stable. They've even been found
riding the shock wave of a supernova explosion!
PAHs are also found in meteorites called "carbonaceous chondrites".
These space rocks contain just a little carbon - about 3% by weight.
But, 80% of this carbon is in the form of PAHs.
Here's an interview with a scientist who thinks PAHs were important
precursors of life on Earth:
5) Aromatic world, interview with Pascale Ehrenfreund,
Astrobiology Magazine, available at
http://www.astrobio.net/news/modules.php?op=modload&name=News&file=article&sid=1992
And here's a book she wrote, with a chapter on organic molecules
in space:
6) Pascale Ehrenfreud, editor, Astrobiology: Future Perspectives,
Springer Verlag, 2004.
Harold Butner also told me about dust disks that have been seen around
the nearby stars Vega and Epsilon Eridani. By examining these disks,
we may learn about planets and comets orbiting these stars. Comets
emit a lot of dust, and planets affect its motion.
Mathematicians will be happy to know that *symplectic geometry*
is required to simulate the motion of this dust:
7) A. T. Deller and S. T. Maddison, Numerical modelling of
dusty debris disks, Astrophys. J. 625 (2005), 398-413.
Also available as arXiv:astro-ph/0502135
Okay... now for a bit about Hochschild cohomology. I want to
outline a conceptual proof of Deligne's conjecture that the
Hochschild cochain complex is an algebra for the little 2-cubes
operad. There are a bunch of proofs of this by now. Here's a
great introduction to the story:
8) Maxim Kontsevich, Operads and motives in deformation
quantization, available as arXiv:math/9904055.
I was inspired to seek a more conceptual proof by some conversations
I had with Simon Willerton in Sheffield this summer, and this paper
of his:
9) Andrei Calderou and Simon Willerton, The Mukai pairing, I:
a categorical approach, available as arXiv:0707.2052
But, while trying to write up a sketch of this more conceptual
proof, I discovered that it had already been worked out:
10) P. Hu, H. Kriz and A. A. Voronov, On Kontsevich's Hochschild
cohomology conjecture, available at arXiv:math.AT/0309369.
This was a bit of a disappointment - but also a relief. It
means I don't need to worry about the technical details: you
can just look them up! Instead, I can focus on sketching the
picture I had in mind.
If you don't know anything about Hochschild cohomology, don't worry!
It only comes in at the very end. In fact, the conjecture
follows from something simpler and more general. So, what you
really need is a high tolerance for category theory, homological
algebra and operads.
First, suppose we have any monoidal category. Such a category
has a tensor product and a unit object, which we'll call I. Let
end(I) be the set of all endomorphisms of this unit object.
Given two such endomorphisms, say
f: I -> I
and
g: I -> I
we can compose them, getting
f o g: I -> I
This makes end(I) into a monoid. But we can also tensor f and
g, and since I tensor I is isomorphic to I in a specified way,
we can write the result simply as
f tensor g: I -> I
This makes end(I) into a monoid in another, seemingly different
way.
Luckily, there's a thing called the Eckmann-Hilton argument which
says these two ways are equal. It also says that end(I) is a
*commutative* monoid! It's easiest to understand this argument
if we write f o g vertically, like this:
f
g
and f tensor g horizontally, like this:
f g
Then the Eckmann-Hilton argument goes as follows:
f 1 f g 1 g
= = g f = =
g g 1 1 f f
Here 1 means the identity morphism 1: I -> I. Each step in the
argument follows from standard stuff about monoidal categories.
In particular, an expression like
f g
h k
is well-defined, thanks to the interchange law
(f tensor g) o (h tensor k) = (f o h) tensor (g o k)
If we want to show off, we can say the interchange law says we've got
a "monoid in the category of monoids" - and the Eckmann-Hilton
argument shows this is just a monoid. See "week100" for more.
But the cool part about the Eckmann-Hilton argument is that we're
just moving f and g around each other. So, this argument has a
topological flavor! Indeed, it was first presented as an argument
for why the second homotopy group is commutative. It's all about
sliding around little rectangles... or as we'll soon call
them, "little 2-cubes".
Next, let's consider a version of this argument that holds only
"up to homotopy". This will apply when we have not a *set*
of morphisms from any object X to any object Y, but a *chain
complex* of morphisms.
Instead of getting a set end(I) that's a commutative monoid, we'll
get a cochain complex END(I) that's a commutative monoid "up to
coherent homotopy". This means that the associative and commutative
laws hold up to chain homotopies, which satisfy their own laws up to
homotopy, ad infinitum.
More precisely, END(I) will be an "algebra of the little 2-cubes
operad". This implies that for every configuration of n little
rectangles in a square:
---------------------
| |
| ----- |
| ----- | | |
|| | | | |
|| | | | |
| ----- | | |
| ----- |
| ---------------- |
| | | |
| ---------------- |
| |
---------------------
we get an n-ary operation on END(I). For every homotopy between
such configurations:
--------------------- ---------------------
| | | ----- |
| ----- | || | ---- |
| ----- | | | || | | | |
|| | | | | || | | | |
|| | | | | || | | | |
| ----- | | | ---> | ----- | | |
| ----- | | ----- |
| ---------------- | | ------- |
| | | | | | | |
| ---------------- | | ------- |
| | | |
--------------------- ---------------------
we get a chain homotopy between n-ary operations on END(I). And
so on, ad infinitum.
For more on the little 2-cubes operad, see "week220". In fact,
what I'm trying to do now is understand some mysteries I described
in that article: weird relationships between the little 2-cubes
operad and Poisson algebras.
But never mind that stuff now. For now, let's see how easy it is to
find situations where there's a chain complex of morphisms between
objects. It happens throughout homological algebra!
If that sounds scary, you should refer to a book like this as you
read on:
10) Charles Weibel, An Introduction to Homological Algebra,
Cambridge U. Press, Cambridge, 1994.
Okay. First, suppose we have an abelian category. This provides a
context in which we can reason about chain complexes and cochain
complexes of objects. A great example is the category of R-modules
for some ring R.
Next, suppose every object X in our abelian category has an
"projective resolution" - that is, a chain complex
d_0 d_1 d_2
X_0 <--- X_1 <--- X_2 <--- ...
where each guy X_i is projective, and the homology groups
ker (d_i)
H_i = -------------
im (d_{i-1})
are zero except for H_0, which equals X. You should think of
a projective resolution as a "puffed-up" version of X that's
better for mapping out of than X itself.
Given this, besides the usual set hom(X,Y) of morphisms from the
object X to the object Y, we also get a cochain complex which I'll
call the "puffed-up hom":
HOM(X,Y)
How does this work? Simple: replace X by a chosen projective
resolution
X^0 <--- X^1 <--- X^2 <--- ...
and then map this whole thing to Y, getting a cochain complex
hom(X^0,Y) ---> hom(X^1,Y) ---> hom(X^2,Y) ---> ...
This chain complex is the puffed-up hom, HOM(X,Y).
Now, you might hope that the puffed-up hom gives us a new category
where the hom-sets are actually cochain complexes. This is morally
true, but the composition
o: HOM(X,Y) x HOM(Y,Z) -> HOM(X,Z)
probably isn't associative "on the nose". However, I think it should
be associative up to homotopy! This homotopy probably won't satisfy
the law you'd hope for - the pentagon identity. But, it should
satisfy the pentagon identity up to homotopy! In fact, this should
go on forever, which is what we mean by "up to coherent homotopy".
This kind of situation is described by an infinite sequence of shapes
called "associahedra" discovered by Stasheff (see "week144").
If this is the case, instead of a category we get an "A-infinity
category": a gadget where the hom-sets are cochain complexes and the
associative law holds up to coherent homotopy. I'm not sure the
puffed-up hom gives an A-infinity category, but let's assume so and
march on.
Suppose we take any object X in our abelian category. Then we get
a cochain complex
END(X) = HOM(X,X)
equipped with a product that's associative up to coherent homotopy.
Such a thing is known as an "A-infinity algebra". It's just an
A-infinity category with a single object, namely X.
Next suppose our abelian category is monoidal. (To get the tensor
product to play nice with the hom, assume tensoring with any object
is right exact.) Let's see what happens to the Eckmann-Hilton
argument. We should get a version that holds "up to coherent
homotopy".
Let I be the unit object, as before. In addition to composition:
o: END(I) x END(I) -> END(I)
tensoring should give us another product:
tensor: END(I) x END(I) -> END(I)
which is also associative up to coherent homotopy. So, END(I) should
be an A-infinity algebra in two ways. But, since composition
and tensoring in our original category get along nicely:
(f tensor g) o (h tensor k) = (f o h) tensor (g o k)
END(I) should really be an A-infinity algebra in the category of
A-infinity algebras!
Given this, we're almost done. A monoid in the category of monoids
is a commutative monoid - that's another way of stating what the
Eckmann-Hilton argument proves. Similarly, an A-infinity algebra in
the category of A-infinity algebras is an algebra of the little
2-cubes operad. So, END(I) is an algebra of the little 2-cubes
operad.
Now look at an example. Fix some algebra A, and take our
monoidal abelian category to have:
A-A bimodules as objects
A-A bimodule homomorphisms as morphisms
Here the tensor product is the usual tensor product of bimodules,
and the unit object I is A itself. And, as Simon Willerton pointed
out to me, END(I) is a chain complex whose homology is familiar:
it's the "Hochschild homology" of A.
So, the cochain complex for Hochschild cohomology is an algebra of
the little 2-cubes operad! But, we've seen this as a consequence
of a much more general fact.
To wrap up, here are a few of the many technical details I glossed
over above.
First, I said a projective resolution of X is a puffed-up version of
X that's better for mapping out of. This idea is made precise
in the theory of model categories. But, instead of calling it a
"puffed-up version" of X, they call it a "cofibrant replacement" for
X. Similarly, a puffed-up version of X that's better for mapping
into is called a "fibrant replacement".
For a good introduction to this, try:
11) Mark Hovey, Model Categories, American Mathematical Society,
Providence, Rhode Island, 1999.
Second, I guessed that for any abelian category where every object has
a projective resolution, we can create an A-infinity category using
the puffed-up hom, HOM(X,Y). Alas, I'm not really sure this is true.
Hu, Kriz and Voronov consider a more general situation, but what I'm
calling the "puffed-up hom" should be a special case of their "derived
function complex". However, they don't seem to say what weakened sort
of category you get using this derived function complex - maybe an
A-infinity category, or something equivalent like a quasicategory or
Segal category? They somehow sidestep this issue, but to me it's
interesting in its own right.
At this point I should mention something well-known that's similar
to what I've been talking about. I've been talking about the
"puffed-up hom" for an abelian category with enough projectives.
But most people talk about "Ext", which is the cohomology of the
puffed-up hom:
Ext^i(X,Y) = H^i(HOM(X,Y))
And, while I want
END(X) = HOM(X,X)
to be an A-infinity algebra, most people seem happy to have
Ext(X) = H(HOM(X,X))
be an A-infinity algebra. Here's a reference:
12) D.-M. Lu, J. H. Palmieri, Q.-S. Wu and J. J. Zhang,
A-infinity structure on Ext-algebras, available as
arXiv:math.KT/0606144.
I hope they're secretly getting this A-infinity structure on
H(HOM(X,X)) from an A-infinity structure on HOM(X,X). They don't
come out and say this is what they're doing, but one promising
sign is that they use a theorem of Kadeishvili, which says that
the cohomology of an A-infinity algebra is an A-infinity algebra.
Finally, the really interesting part: how do we make an A-infinity
algebra in the category of A-infinity algebras into an algebra of
the little 2-cubes operad? This is the heart of the "homotopy
Eckmann-Hilton argument".
I explained operads, and especially the little k-cubes operad,
back in "week220". The little k-cubes operad is an operad in
the world of topological spaces. It has one abstract n-ary operation
for each way of sticking n little k-dimensional cubes in a big
one, like this:
---------------------
| |
| ----- |
| ----- | | |
|| | | | |
|| | | | | typical
| ----- | | | 3-ary operation in the
| ----- | little 2-cubes operad
| ---------------- |
| | | |
| ---------------- |
| |
---------------------
A space is called an "algebra" of this operad if these abstract
n-ary operations are realized as actual n-ary operations on the
space in a consistent way. But, when we study the homology
of topological spaces, we learn that any space gives a chain complex.
This let's us convert any operad in the world of topological spaces
into an operad in the world of chain complexes. Using this, it also
makes sense to speak of a *chain complex* being an algebra of the
little k-cubes operad. For that matter, a cochain complex.
Let's use "E(k)" to mean the chain complex version of the little
k-cubes operad.
An "A-infinity algebra" is an algebra of a certain operad called
A-infinity. This isn't quite the same as the operad E(1), but it's
so close that we can safely ignore the difference here: it's
"weakly equivalent".
Say we have an A-infinity algebra in the category of A-infinity
algebras. How do we get an algebra of the little 2-cubes operad,
E(2)?
Well, there's a way to tensor operads, such that an algebra of
P tensor Q is the same as a P-algebra in the category of Q-algebras.
So, an A-infinity algebra in the category of A-infinity algebras is
the same as an algebra of
A-infinity tensor A-infinity
Since A-infinity and E(1) are weakly equivalent, we can turn this
algebra into an algebra of
E(1) tensor E(1)
But there's also an obvious operad map
E(1) tensor E(1) -> E(2)
since the product of two little 1-cubes is a little 2-cube.
This too is a weak equivalence, so we can turn our algebra of
E(1) tensor E(1) into an algebra of E(2).
The hard part in all this is showing that the operad map
E(1) tensor E(1) -> E(2)
is a weak equivalence. In fact, quite generally, the map
E(k) tensor E(k') -> E(k+k')
is a weak equivalence. This is Proposition 2 in the paper by
Hu, Kriz and Voronov, based on an argument by Gerald Dunn:
13) Gerald Dunn, Tensor products of operads and iterated loop
spaces, Jour. Pure Appl. Alg 50 (1988), 237-258.
Using this, they do much more than what I've sketched: they
prove a conjecture of Kontsevich which says that the Hochschild
complex of an algebra of the little k-cubes operad is an algebra
of the little (k+1)-cubes operad!
That's all for now. Sometime I should tell you how this is related
to Poisson algebras, 2d TQFTs, and much more. But for now, you'll
have to read that in Kontsevich's very nice paper.
-----------------------------------------------------------------------
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