- #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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mathematics and physics, as well as some of my research papers, can be

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http://math.ucr.edu/home/baez/twfcontents.html

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http://math.ucr.edu/home/baez/twfshort.html

If you just want the latest issue, go to

http://math.ucr.edu/home/baez/this.week.html

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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