November 26, 2007

This Week’s Finds in Mathematical Physics (Week 258)

Posted by John Baez

In week258 of This Week’s Finds, learn what happens when sand grains flow. Read more about dust in the Red Rectangle:

and discover why some of it looks like this:

Then, find out about Deligne’s conjecture concerning Hochschild cohomology — and get a nice workout in homological algebra, categories and operads.

Café regulars will be pleased to know that the proof of Deligne’s conjecture is really just an infinitely categorified version of the Eckmann–Hilton argument!

Others may be pleased to know that I explain what that means, in terms so simple that any math PhD who studied homological algebra can understand it.

Posted at November 26, 2007 4:12 AM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 258)

a nice workout in homological algebra, categories and operads.

Some more or less random questions and/or remarks, all spiraling in more or less on what you have been talking about in this TWF:

So let $C$ be our abelian category.

The Dold-Kan correspondence says that we have an equivalence of categories $\mathrm{Simp}\left(C\right)\simeq {\mathrm{Ch}}_{•}\left(C\right)$ between simplicial objects in $C$ and non-negatively graded chain complexes in $C$. The direction $\mathrm{Simp}\left(C\right)\to {\mathrm{Ch}}_{•}\left(C\right)$ is forming the “normalized chain complex” which corresponds to throwing away lots of face maps except for one at each level, restricting that to the kernel of some of the others, and regarding it as the differential in our complex.

Now, first question: how precisely do I connect this to higher categories now?

I want to combine this with the nerve functor $\omega \mathrm{Cat}\left(C\right)\stackrel{\mid \cdot \mid }{\to }\mathrm{Simp}\left(C\right)$ which sends an $\omega$-category internal to $C$ to its simplicial nerve. I want to get an equivalence $\omega \mathrm{Cat}\left(C\right)\simeq {\mathrm{Ch}}_{•}\left(C\right)$ this way. Do I? If not, can you fix this?

This was the first question. Now comes the first big mystery™:

$\omega$-categories correspond to non-negatively graded chain complexes (and nicely so: in degree $k$ of the chain complex we have the space of $k$-morphisms starting at the 0 $\left(k-1\right)$-morphism, roughly).

These naturally sit in arbitrarily graded chain complexes. In particular, a couple of nice and natural operations on non-negatively graded chain complexes will produce not-necessarily-non-negatively graded chain complexes from them.

Most notably: forming the dual chain complex and forming the internal hom from that.

So: in the context that $\omega$-categories correspond to non-negatively graded chain complexes, what corresponds to positively graded chain complexes?

Maybe $\omega$-co-categories??

Is there a general way in which the “dual” of a category would be a co-category? (Probably like the “dual” of a group, namely functions on the group elements, has a co-algebra structure on it.) (And if so, preferably with $\omega$ everywhere.)

I have Weibel’s book sitting here (Danny’s copy, to be honest) and need to refresh my memory about a bunch of things. It’s been a while.

I’d be most pleased if somebody could give an exegesis of the standard homological algebra constructions entirely in terms of $\omega \mathrm{Cat}\left(C\right)$, as far as possible. I sense in John’s TWF the tendency to do just that.

It seems to me that I’d need such an interpretation in order to understand what some easy and obvious constructions in homological algebra mean.

For instance, looking at thet AKSZ paper one finds: they say we should look at two non-negatively graded cochain complexes (parameter space $\mathrm{par}$ and target space $\mathrm{tar}$), but then form the space of fields (the hom object between these two) in the context of arbitrarily graded complexes, hence allowing the complex $\mathrm{hom}\left(\mathrm{par},\mathrm{tar}\right)$ to be positively and negatively graded.

(positive = ghosts, negative = anti-xyz).

It’s a simple construction in the homological algebra context. But its true interpretation is quite mysterious (currently to me): both $\mathrm{par}$ and $\mathrm{tar}$ I know come from certain higher categories. But $\mathrm{hom}\left(\mathrm{par},\mathrm{tar}\right)$ with negative degrees allowed does not – at least not in the same obvious manner.

So what’s going on??

Posted by: Urs Schreiber on November 26, 2007 6:49 PM | Permalink | Reply to this

dual categories and co-categories

Let’s see. Parts of my questions have an obvious answer. Let me try:

Let $V$ be a closed monoidal category such that we can enrich over it. Assume in addition that $V$ has duals on objects.

Then for $C$ any $V$-category with composition ${\circ }_{x,y,z}:\mathrm{Hom}\left(x,y\right)\otimes \mathrm{Hom}\left(y,z\right)\to \mathrm{Hom}\left(x,z\right)$

we obtain a dual co-catgeory enriched over $V$ simply by hitting everything in sight with the duality operation. So we get a co-composition ${\circ }_{x,y,z}^{*}:\mathrm{Hom}\left(x,z{\right)}^{*}\to \mathrm{Hom}\left(x,y{\right)}^{*}\otimes \mathrm{Hom}\left(y,z{\right)}^{*}$ obviously.

Hm. And I guess if I do everything internal to $V$, the same general statement is true. And if I then imagine I have an $\omega$-category internal to $V$, the same kind of statement is still true.

That was easy enough.

So next suppose that we have one $\omega$-category acting on another. Say $G$ is an $\omega$-category with a single object and we have a morphism

$G\to \mathrm{End}\left(C\right)\phantom{\rule{thinmathspace}{0ex}}.$

Then, somehow, we want to dualize $G$, probably obtaining a co-$\omega$-category this way , and then merge the result with $C$ to some single new entity.

What might that be?

Posted by: Urs Schreiber on November 26, 2007 7:14 PM | Permalink | Reply to this

Re: dual categories and co-categories

After being too verbose, as usual, I asked

What might that be?

More briefly: in a way I am asking:

What might the action $\omega$-thing of an $\omega$-groupoid acting on an $\omega$-category be?

And I am looking for an answer which, when hit with some big $\mathrm{Lie}$ operation spits out a chain complex with the $\omega$-groupoid in one part of its degrees, and the thing it acts on in the other.

Posted by: Urs Schreiber on November 26, 2007 8:15 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Urs wrote:

Okay, great! You seem to have sensed that “week258” was secretly about $\omega$-categories — or maybe you just read the little hint in this blog entry. I didn’t want to get into that aspect explicitly in “week258”, since it would further clutter what was already an overly jargonesque explanation of what ultimately is a very simple idea.

But, whenever I think about chain complexes, I’m secretly thinking of them as specially nice strict $\omega$-categories.

Okay…

So let $C$ be our abelian category.

The Dold-Kan correspondence says that we have an equivalence of categories $\mathrm{Simp}\left(C\right)\simeq {\mathrm{Ch}}_{•}\left(C\right)$ between simplicial objects in $C$ and non-negatively graded chain complexes in $C$. The direction $\mathrm{Simp}\left(C\right)\to {\mathrm{Ch}}_{•}\left(C\right)$ is forming the “normalized chain complex” which corresponds to throwing away lots of face maps except for one at each level, restricting that to the kernel of some of the others, and regarding it as the differential in our complex.

Right.

Now, first question: how precisely do I connect this to higher categories now?

I want to combine this with the nerve functor $\omega \mathrm{Cat}\left(C\right)\stackrel{\mid \cdot \mid }{\to }\mathrm{Simp}\left(C\right)$ which sends an $\omega$-category internal to $C$ to its simplicial nerve. I want to get an equivalence $\omega \mathrm{Cat}\left(C\right)\simeq {\mathrm{Ch}}_{•}\left(C\right)$ this way. Do I?

Yes you do! Here I’m assuming you mean strict $\omega$-categories.

You can use a nerve construction as you outline. But, it’s equally easy to see directly that the category

• $\omega \mathrm{Cat}\left(C\right)$: strict $\omega$-categories internal to the abelian category $C$, with strict $\omega$-functors as morphisms.

is equivalent to the category

• ${\mathrm{Ch}}_{•}\left(C\right)$: nonnegatively graded chain complexes in $C$, with chain maps as morphisms.

It’s fun and instructive to build this equivalence ‘by hand’, reinterpreting globes in terms of chains. But if you’re too busy, there’s a paper by Brown and Higgins that does it:

• Ronald Brown and Philip J. Higgins, Cubical abelian groups with connections are equivalent to chain complexes.

Abstract: The theorem of the title is deduced from the equivalence between crossed complexes and cubical $\omega$-groupoids with connections proved by the authors in 1981. In fact we prove the equivalence of five categories defined internally to an additive category with kernels.

I believe that three of these five are $\omega \mathrm{Cat}\left(C\right)$, ${\mathrm{Ch}}_{•}\left(C\right)$ and $\mathrm{Simp}\left(C\right)$, while the others have a cubical flavor. But if you prefer octahedra and their higher-dimensional generalizations, you can probably use them too — this idea is incredibly robust!

Since 2-vector spaces are equivalent to 2-term chain complexes, as described in Section 3, it should not be surprising that ${L}_{\infty }$-algebras are related to the categorified Lie algebras we are discussing here. An elegant but rather highbrow way to approach this is to use the theory of operads [MSS]. An ${L}_{\infty }$-algebra is actually an algebra of a certain operad in the symmetric monoidal category of chain complexes, called the ‘${L}_{\infty }$ operad’. Just as categories in $Vect$ are equivalent to 2-term chain complexes, strict $\omega$-categories in $Vect$ can be shown equivalent to general chain complexes, by a similar argument [BH]. Using this equivalence, we can transfer the ${L}_{\infty }$ operad from the world of chain complexes to the world of strict $\omega$-category objects in $Vect$, and define a semistrict Lie $\omega$-algebra to be an algebra of the resulting operad.

I also sketched how to see a chain complex in $C$ as a strict $\omega$-category in $C$ starting at the bottom of page 18 here:

This is getting long, so I’ll quit this comment here and talk about other things in another comment. Your question:

So: in the context that $\omega$-categories correspond to non-negatively graded chain complexes, what corresponds to positively graded chain complexes?

was very much on my mind while writing “week258”. Or at least it would have been if you meant to write “what corresponds to non-positively graded chain complexes?” Is that what you meant?

Posted by: John Baez on November 26, 2007 7:36 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Brown and Higgins wrote:

In fact we prove the equivalence of five categories defined internally to an additive category with kernels.

I wrote:

I believe that three of these five are $\omega \mathrm{Cat}\left(C\right)$, ${\mathrm{Ch}}_{•}\left(C\right)$ and $\mathrm{Simp}\left(C\right)$, while the others have a cubical flavor.

Actually they don’t do $\mathrm{Simp}\left(C\right)$ — I guess the Dold-Kan theorem was too well-known to bother stating! They show these five are equivalent:

1. chain complexes in $C$
2. crossed complexes in $C$
3. cubical sets with connections in $C$
4. cubical $\omega$-groupoids with connections in $C$
5. globular strict $\omega$-groupoids in $C$

It’s easy to see that globular strict $\omega$-categories in $C$ are the same as globular strict $\omega$-groupoids in $C$. So, item 5 is just what you’re calling $\omega \mathrm{Cat}\left(C\right)$. And, of course, item 1 is your ${\mathrm{Ch}}_{•}\left(C\right)$.

Posted by: John Baez on November 26, 2007 7:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

You seem to have sensed that “week258” was secretly about $\omega$-categories

Well, it almost seemed like you were responding to my questions here :-)

Or at least it would have been if you meant to write “what corresponds to non-positively graded chain complexes?” Is that what you meant?

Yes! These sign issues are a nuisance…

Posted by: Urs Schreiber on November 26, 2007 7:51 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

But if you prefer octahedra and their higher-dimensional generalizations, you can probably use them too

You could help fix yet another gap in my education here:

somehow I understand that Ross Street’s orientals, or probably some generalization of them, is supposed to make precise what you just indicated: a notion of $\omega$-category based on a rather arbitrary geometric form (simplex, cube, etc.)

But I don’t really know the details.

Hm, while googling for this I ran into

Richard Steiner, Orientals

which even relates all that to chain complexes:

We show that the category of orientals is isomorphic to a subcategory of the category of chain complexes.

Hm, this is closely related to what I was talking about with Bruce in Bakewell. I wonder if Bruce is reading this here…

With Bruce I was chatting about this (not that our joint expertise was supposed to overcome the critical mass threshold):

dg-manifolds (linear approximation to smooth $\omega$-groupoids) are modules over the endomorphisms of the odd line. This is a cool way of saying that most of their properties are encoded in the unique law of the universe, which reads ${d}^{2}=0$.

I was looking for an analogous way to describe $\omega$-groupoids themselves. If I think of them as Kan-complexes, hence in particular as simplicial objects, I seem to be getting close: I could try to look at simplicial objects internal to $ℤ$-modules and then encode all face and degeneracy information in something like ${\partial }^{2}=0$ acting on that.

Does anyone see what I am trying to get at here? I’ll see what Richard Steiner has to say about this…

Posted by: Urs Schreiber on November 26, 2007 8:57 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Urs wrote:

Somehow I understand that Ross Street’s orientals, or probably some generalization of them, is supposed to make precise what you just indicated: a notion of $\omega$-category based on a rather arbitrary geometric form (simplex, cube, etc.)

But I don’t really know the details.

The $n$th oriental is the free strict $\omega$-category on an $n$-simplex.

Orientals are Street’s method of solving this problem: what is the simplicial nerve of a strict $\omega$-category?

To answer this sort of problem, we need to take a strict $\omega$-category $C$ and figure out what the $n$-simplices in its nerve are.

An $n$-simplex in $\mathrm{Nerve}\left(C\right)$ is just a map from $\Delta \left[n\right]$ to $\mathrm{Nerve}\left(C\right)$. Here $\Delta \left[n\right]$ is the ‘walking $n$-simplex’ — the simplicial set that consists mainly of an $n$-simplex, but also of course its faces and degeneracies. Most people call this the ‘simplicial $n$-simplex’, but that sounds a bit scary at first.

So, the set of $n$-simplices in $\mathrm{Nerve}\left(C\right)$ is

$\mathrm{hom}\left(\Delta \left[n\right],\mathrm{Nerve}\left(C\right)\right)$

where we’re taking $\mathrm{hom}$ in the world of simplicial sets.

But the way these things work, the $\mathrm{Nerve}$ functor should be right adjoint to some functor — let’s call it ${\Pi }_{\infty }$ — which takes a simplicial set and works out its ‘fundamental $\infty$-category’. So, we should have

$\mathrm{hom}\left(X,\mathrm{Nerve}\left(C\right)\right)\cong \mathrm{hom}\left({\Pi }_{\infty }\left(X\right),C\right)$

and in particular

$\mathrm{hom}\left(\Delta \left[n\right],\mathrm{Nerve}\left(C\right)\right)\cong \mathrm{hom}\left({\Pi }_{\infty }\left(\Delta \left[n\right]\right),C\right)$

So, we’ll understand a lot about the nerve of $C$ if we can understand ${\Pi }_{\infty }\left(\Delta \left[n\right]\right)$ — the fundamental $\omega$-category of the walking $n$-simplex. And, this is what Street calls the $n$th oriental!

If you want to understand this stuff, I urge you to ponder Sreet’s Conspectus of Australian Category Theory, a nice paper which will eventually show up in the volume Peter May and I are editing.

In the section which describes his work on simplicial weak $\omega$-categories, he says:

I decided to concentrate on one aspect of the problem. How do we rigorously define the nerve of an $n$-category? After unsuccessfully looking for an easy way out using multiple categories and multiply simplicial sets (I sent several letters to John Roberts about this), I realized that the problem came down to defining the free $n$-category ${O}_{n}$ on the $n$-simplex. Meaning had to be given to the term ‘free’ in this context: free on what kind of structure? How was an ‘n’-simplex an example of the structure? The structure required was an $n$-computad. The definition of $n$-computad and free $n$-category on an $n$-computad is done simultaneously by induction on $n$ (see St17, Pw1, St21, St22, St29). An element of dimension $n$ of the nerve $NA$ of an $\omega$-category $A$ is an $n$-functor from ${O}_{n}$ to $A$. Things began to click once I drew the following picture of the $4$-simplex:

Click on the picture for a bigger version. It’s really a picture of the oriental ${O}_{4}$, if you know how to look at it correctly. It should remind you of the ‘pentagonator’.

Posted by: John Baez on November 27, 2007 1:45 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

It IS the ur-pentagonator. Just take the tree corresponding to the triangulation. It continues to work for the associahedra. In fact I think ?Street? drew the picture for the 5-simplex.

Posted by: jim stasheff on November 27, 2007 2:35 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

If you read Street’s conspectus, you’ll see he even drew the picture for the 6-simplex — and you’ll see how long it took him!

Posted by: John Baez on November 27, 2007 7:34 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

“ah yes, I remember it well”

It’s similar pictures leading to Gordon, Power and Street that led me to realize that my former colleague Gordon and I had more in common than friendship.

Posted by: jim stasheff on November 28, 2007 1:55 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Urs meant to write:

So: in the context that $\omega$-categories correspond to non-negatively graded chain complexes, what corresponds to non-positively graded chain complexes?

Maybe $\omega$-co-categories??

One way to tackle this uses the fact that whenever $C$ is an abelian category, so is ${C}^{\mathrm{op}}$. The people who invented this concept cleverly made the definition self-dual!

So, here goes:

First, a non-positively graded chain complex in some abelian category $C$ is the same as a non-negatively graded cochain complex in $C$.

Second, a non-negatively graded cochain complex in $C$ is the same as a non-negatively graded chain complex in ${C}^{\mathrm{op}}$.

Third, a non-negatively graded chain complex in ${C}^{\mathrm{op}}$ is the same as a strict $\omega$-category in ${C}^{\mathrm{op}}$.

So, a non-positively graded chain complex in $C$ is the same as a strict $\omega$-category in ${C}^{\mathrm{op}}$.

I suppose you could call this an ‘$\omega$-co-category’ in $C$, but fewer people would understand you — like, 3 instead of 10.

However, we don’t always want to separate the positive and negative parts of a chain complex and treat them separately! What if we’ve got a full-fledged, $ℤ$-graded chain complex in $C$?

This gives a strict $ℤ$-category in $C$!

A $ℤ$-category has $n$-morphisms for $n$ going down all the way to $-\infty$, as well as all the up to $+\infty$.

Alas, the only people I know who have discussed $ℤ$-categories are James Dolan and myself. And, we haven’t written too much about them — just a few remarks in various papers.

It’s not hard to define strict $ℤ$-categories, internalize them, and prove that a $ℤ$-graded chain complex in $C$ is the same as a strict $ℤ$-category in $C$.

The trickier part is to make sure that a strict $ℤ$-groupoid works out to be precisely a $ℤ$-graded chain complex of abelian groups. You need to make sure the Eckmann–Hilton theorem kicks in and makes everything abelian.

The really tricky thing would be to define weak $ℤ$-groupoids in a purely algebraic way, and then make sure they’re the same as spectra. In the field of ‘stable homotopy theory’, algebraic topologists use spectra as a generalization of chain complexes in precisely this way. The word ‘stable’ means ‘with homotopy groups going infinitely far down’. But, these guys don’t define spectra starting from a theory of weak $ℤ$-categories. Not yet, anyway.

There’s a lot more to say about this… Most things about weak $ℤ$-categories would be hard to make precise at present. But, many things involving strict $ℤ$-categories should be quite easy to make precise.

Posted by: John Baez on November 26, 2007 10:05 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

However, we don’t always want to separate the positive and negative parts of a chain complex and treat them separately! What if we’ve got a full-fledged, $ℤ$-graded chain complex in $C$?

Exactly, that’s what I keep asking. Starting here, continuing here and culminating here.

Back then I did receive a couple of helpful replies to this by David Ben-Zvi here, here and here.

I found these helpful, but wanted more: I wanted a conceptual understanding of what it means, $\omega$-groupoid-wise, when I start with my Lie $\omega$-groupoid, turn it into a non-negatively graded chain complex by linearizing it – and then adding stuff in the negative degrees.

In On Lie $\infty$-modules and the BV complex I am starting to argue that having positive and negative degrees is the hallmark of having one $\omega$-thing acting on another $\omega$-thing.

This crucially involves some of the observations you mentioned, like

a non-positively graded chain complex in some abelian category $C$ is the same as a non-negatively graded cochain complex in $C$.

Namely, when we form the Lie-Rinehart $\infty$-pair with one Lie $\infty$-algebra $g$ (a chain complex, in particular!) acting on some module $B$ (another chain complex!) this structure is encoded in the correspondonding Chevalley-Eilenberg algebra, which is defined on the complex

${g}^{*}\oplus B\phantom{\rule{thinmathspace}{0ex}}.$

Notice that, due to the fact that $g$ here has been dualized, this is now a complex in arbitrary degree. And it’s a fact that equipping that complex with a dg-algebra structure encodes

- both the Lie $\infty$-algebra structure of $g$

- as well as its action on $B$.

Therefore my conclusion was:

arbitrarily graded chain complexes (at least when carrying a dg-algebra structure) have to do with actions of $\omega$-categories on each other.

But luckily you now provide the kind of interpretation I was hoping you would give:

What if we’ve got a full-fledged, $ℤ$-graded chain complex in $C$?

This gives a strict $ℤ$-category in $C$!

Ah!

Clearly my next question would be for references:

Alas, the only people I know who have discussed $ℤ$-categories are James Dolan and myself. And, we haven’t written too much about them — just a few remarks in various papers.

I guess I can imagine how to define them. What I would really like to know is:

- how are you thinking about $ℤ$-categories in the grand scheme of things?

- where do you apply them?

- do you at all see a connection with the situations in which you have been thinking about $ℤ$-categories and the situation I sketched, where we are dealing with actions of one $\omega$-category on another?

Posted by: Urs Schreiber on November 27, 2007 9:51 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Urs wrote:

I guess I can imagine how to define them. What I would really like to know is:

- how are you thinking about $ℤ$-categories in the grand scheme of things?

They’re all about ‘stabilization’ — the process that occurs as you march down the periodic table.

A $k$-tuply monoidal $n$-category should be the same as a $\left(k+1\right)$-tuply monoidal $n$-category when $k$ gets big enough, I guess around $k=2n+2$. So at this point we can just call it a stable $n$-category.

But, a stable $n$-category is a special sort of stable $\left(n+1\right)$-category: one with only identity morphisms. Conversely, we should be able to decategorify a stable $\left(n+1\right)$-category and get a stable $n$-category. This suggests that we define a stable $\infty$-category to be an $\infty$-category equipped with extra structure such that whenever you decategorify it to get an $n$-category for any finite $n$, the result is stable.

You can think of a stable $\infty$-category as having $n$-morphisms for each $n\ge 0$, together with an ‘infinitely deep basement’ consisting of one identity $n$-morphism for each $n<0$.

However, if you think about it this way, the indexing is a bit arbitrary. Why require that the interesting $n$-morphisms show up only for $n>0$? We can reindex them so they show up for $n>k$ for any $k\in ℤ$, even negative $k$.

At this point we’re tempted to consider $ℤ$-categories — gadgets that can have nontrivial $n$-morphisms for all $n\in ℤ$. But, we want to retain the benefits of stability. So, we should define them carefully. I could explain how, but I prefer to let you ponder it a bit first, to see why it requires care.

The key point is: algebraic topologists have been thinking about this stuff for many decades already. However, instead of $\infty$-categories, they think about $\infty$-groupoids, which they call ‘spaces’. They call stable $\infty$-groupoids ‘infinite loop spaces’. And, they call $ℤ$-groupoids ‘spectra’.

So, many of the clever tricks we need to perform can simply be stolen from them!

In particular, this business about needing to define $ℤ$-categories ‘carefully’ is very famous in the theory of spectra. It took algebraic topologists a long time to reach their current understanding of spectra.

- where do you apply them?

$ℤ$-categories are very important: they show up whenever algebraic topologists study stable homotopy theory — meaning things like $ℤ$-graded chain complexes (for babies) or spectra (for grownups). A spectrum is just a weak $ℤ$-groupoid. A chain complex is just a strict $ℤ$-groupoid!

Our job is simply to stop saying ‘groupoid’ all the time, and say ‘category’ instead.

- do you at all see a connection with the situations in which you have been thinking about $ℤ$-categories and the situation I sketched, where we are dealing with actions of one $\omega$-category on another?

I don’t have time for a thoughtful answer now, but place $ℤ$-categories show up is this: if $X$ and $Y$ are nonnegatively graded chain complexes, the sensible notion of $\mathrm{hom}\left(X,Y\right)$ is $ℤ$-graded! The negative grades creep in even if you hadn’t wanted them. This is how cochain complexes snuck into “week258”.

And of course homs are closely related to actions…

Posted by: John Baez on November 27, 2007 8:00 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Just remind me: when you say $\infty$-groupoid, what precisely do you mean? Kan complex, quasi-category, $\omega$-groupoid, complicial set, …? And does it matter?

but one place $ℤ$-categories show up is this: if $X$ and $Y$ are nonnegatively graded chain complexes, the sensible notion of $\mathrm{hom}\left(X,Y\right)$ is \mathbb{Z}-graded! The negative grades creep in even if you hadn’t wanted them.

Yes, that’s precisely the reason why I am trying to straighten this out:

AKSZ show say that when you quantize a thing that looks like $X$ and propagates on $Y$, with $X$ and $Y$ $\infty$-groupoids of sorts, then the space of fields should be taken as the $\mathrm{hom}\left(X,Y\right)$ of $ℤ$-categories! Instead of what you’d rather expect: the hom in $\infty$-groupoids.

+) The positive degrees of $\mathrm{hom}\left(X,Y\right)$ are the ghosts.

0) The zero degrees of $\mathrm{hom}\left(X,Y\right)$ are the fields.

-) The negative degrees of $\mathrm{hom}\left(X,Y\right)$ are the anti-fields and anti-ghosts.

I could make my life easy and simply accept this as a fact, happily computing in this context. I understand this quite well operationally and it is an extremely cool fact - once you accept it.

But it bothers me. Even if I accept the fact that I need to consider $ℤ$-categories, it’s still strange:

why don’t we take $X$ and $Y$ to be $ℤ$-categories themselves? Probably we could. But then: what does it mean if we don’t?

Posted by: Urs Schreiber on November 27, 2007 10:38 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Urs wrote:

Just remind me: when you say $\infty$-groupoid, what precisely do you mean? Kan complex, quasi-category, $\omega$-groupoid, complicial set,..? And does it matter?

There’s a big difference between strict and weak $\infty$-grouopids, of course.

By a ‘strict $\infty$-groupoid’ I mean a strict globular $\omega$-category with all $n$-morphisms ($n>0$) strictly invertible. Maybe this is what you’re calling an $\omega$-groupoid.

(I don’t particularly like the use of ‘$\omega$-’ to mean ‘strict $\infty$-‘, but perhaps it’s entrenched enough that we should go with it.)

There are lots of different models of weak $\infty$-groupoids that have all been proved equivalent, at least at the level of model categories. That is, there are a bunch of model categories which are all Quillen equivalent, and I’m perfectly happy to call an object of any of these a weak $\infty$-groupoid.

The two most obvious choices are:

• simplicial sets (with Kan complexes as the fibrant objects, so we may use just Kan complexes)
• nice topological spaces (let us say compactly generated weak Hausdorff spaces)

Quasicategories are not at all intended to be a model of weak $\infty$-groupoids; they’re a model of weak $\left(\infty ,1\right)$-categories, which are significantly more general! There are lots of other models of weak $\left(\infty ,1\right)$-categories, and any of these can be restricted to give a model of $\infty$-groupoids, by demanding that the 1-morphisms are weakly invertible.

Complicial sets are also not a model for weak $\infty$-groupoids. They’re a simplicial formulation of strict $\infty$-categories.

Posted by: John Baez on November 28, 2007 7:05 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Okay, thanks a lot.

So this was a really important piece of information for me:

They call stable $\infty$-groupoids ‘infinite loop spaces’. And, they call $ℤ$-groupoids ‘spectra’. #

even though I gather it is maybe supposed to be more or less a tautology.

If I understand correctly, the point is this: an infinite loop space is a space which allows to apply $B$ to it arbitrarily often. But we can apply $B$ $n$ times to things that are $n$-tuply monoidal. So infinite loop spaces correspond to $\infty$-tuply monoidal things. And another word for that is stable things.

Then I also get the point about spectra and $ℤ$-categories, I guess.

Hm, now suppose we’d considered ring spectra. These would correspond to monoidal $ℤ$-categories.

But… Hm. There shouldn’t be anything preventing us from considering monoidal $ℤ$-categories, then double monoidal $ℤ$-categories, then $k$-tuply monoidal $ℤ$-categories.

Then stable $ℤ$-categories??

Suppose I were not just saying these words but could actually handle this: would I just go in a circle and stay withing $ℤ$-categories – or would I arrive at, say, ${ℤ}^{2}$-categories, or maybe ${ℤ}^{ℤ}$-categories or…

Somehow dg-algebras are supposed to be equivalent to (hope I get this right) modules over Eilenberg-MacLane ring spectra? Or something like that, I am probably misremembering what somebody told me over lunch. But maybe I need to better understand that statement. And what it means in terms of $ℤ$-categories.

Posted by: Urs Schreiber on November 28, 2007 7:25 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Urs wrote:

If I understand correctly, the point is this: an infinite loop space is a space which allows to apply B to it arbitrarily often. But we can apply B $n$ times to things that are $n$-tuply monoidal. So infinite loop spaces correspond to $\infty$-tuply monoidal things. And another word for that is stable things.

You’re almost exactly right. James Dolan and I discuss this starting on page 21 of Categorification. It’s standard material in topology; what we do is sketch how to talk about it in the language of $n$-category theory.

But, you may enjoy this slight correction: there’s a difference between a $k$-tuply monoidal $\infty$-groupoid and a $k$-tuply groupal $\infty$-groupoid. In the latter, the $k$ extra multiplication operations have weak inverses.

In topology, a $k$-tuply monoidal $\infty$-groupoid corresponds to an ${E}_{k}$ space. (To be precise, an ${E}_{k}$ space is a space that’s an algebra of the little $k$-cubes operad.)

Among the ${E}_{k}$ spaces are the $k$-fold loop spaces. These are the ones that correspond to $k$-tuply groupal $\infty$-groupoids.

It’s easy to exhibit lots of very interesting ${E}_{k}$ spaces that that aren’t $k$-fold loop spaces. The easiest one is the ‘free ${E}_{k}$ space on a point’. For details, see page 27 of our paper.

Then I also get the point about spectra and $ℤ$-categories, I guess.

You probably do. But: spectra correspond to $ℤ$-groupoids.

I don’t actually know truly interesting examples of $ℤ$-categories that aren’t $ℤ$-groupoids. (By ‘truly interesting’, I mean ones that aren’t just shifted $\infty$-categories.) There must be lots of fun examples; I just haven’t tried to find any.

Hm, now suppose we’d considered ring spectra. These would correspond to monoidal $ℤ$-categories.

You mean monoidal $ℤ$-groupoids.

Yes, ring spectra are $ℤ$-groupoids with a new monoidal structure which we should think of as ‘multiplication’, since it distributes over the previous ‘additive’ stable monoidal structure.

But… Hm. There shouldn’t be anything preventing us from considering monoidal $ℤ$-categories, then double monoidal $ℤ$-categories, then $k$-tuply monoidal $ℤ$-categories.

Yes. In the world of topology, $k$-tuply monoidal $ℤ$-groupoids are very important. They’re called ‘${E}_{k}$ ring spectra’.

Then stable $ℤ$-categories??

Stable $ℤ$-groupoids are also very important in topology. They’re called ‘${E}_{\infty }$ ring spectra’. Many of the most popular spectra are ${E}_{\infty }$ ring spectra.

In short; when we replace abelian groups by $ℤ$-groupoids (aka spectra), rings go to ring spectra and commutative rings go to ${E}_{\infty }$ ring spectra.

This set of analogies is the foundation of brave new algebra. People are starting to redo all of algebraic geometry with ${E}_{\infty }$ ring spectra replacing commutative rings! You can think of this as some sort of ‘$ℤ$-groupoidification’ of algebraic geometry.

It’s about time! The basic ideas are not new:

• J. P. May, F. Quinn, N. Ray and J. Tornehave, ${E}_{\infty }$ Ring Spaces and ${E}_{\infty }$ Ring Spectra, Lecture Notes in Mathematics 577, Springer Verlag, Berlin, 1977.

But, important technical advances have only recently made it practical to do algebraic geometry with ${E}_{\infty }$ ring spectra — and this is what makes brave new algebra a hot topic.

Hmm — it’s time for dinner.

Posted by: John Baez on November 29, 2007 5:29 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

somehow dg-algebras are supposed to be equivalent to (hope I get this right) modules over Eilenberg-MacLane ring spectra?

The statement is apparently:

dg-algebras (of $ℤ$-modules i.e. of abelian groups) are equivalent to

monoids in the category of symmetric spectra that are modules over the Eilenberg-MacLance spectrum $Hℤ$.

Since dg-algebras are also monoids in (co)chain complexes I was wondering whether that means that (co)chain complexes themselves are equivcalent to $Hℤ$-modules.

But this we couldn’t figure out.

Posted by: Urs Schreiber on November 29, 2007 5:30 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

John wrote:

but one place $ℤ$-categories show up is this: if $X$ and $Y$ are nonnegatively graded chain complexes, the sensible notion of $\mathrm{hom}\left(X,Y\right)$ is $ℤ$-graded! The negative grades creep in even if you hadn’t wanted them.

And I said:

Yes, that’s precisely the reason why I am trying to straighten this out:

AKSZ show say that when you quantize a thing that looks like $X$ and propagates on $Y$, with $X$ and $X$ $\infty$-groupoids of sorts, then the space of fields should be taken as the $\mathrm{hom}\left(X,Y\right)$ of $ℤ$-categories! Instead of what you’d rather expect: the hom in $\infty$-groupoids.

+) The positive degrees of $\mathrm{hom}\left(X,Y\right)$ are the ghosts.

0) The zero degrees of $\mathrm{hom}\left(X,Y\right)$ are the fields.

-) The negative degrees of $\mathrm{hom}\left(X,Y\right)$ are the anti-fields and anti-ghosts.

I am about to get into that in more detail in my notes. Here is a figure emphasizing this (important, I think) point a little more:

And now good night!

Posted by: Urs Schreiber on November 28, 2007 11:55 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

In search for hints concerning my conjecture that coalgebra structures on positively and negatively graded chain complexes correspond to Lie $\infty$-algebras (in positive degree) together with an $\infty$-vector space (in negative degree) which they act on, I asked John:

- do you at all see a connection with the situations in which you have been thinking about $ℤ$-categories and the situation I sketched, where we are dealing with actions of one $\omega$-category on another?

After giving plenty of thoughtful answers, John replied to this particular question:

I don’t have time for a thoughtful answer now, but place $ℤ$-categories show up is this: if X and Y are nonnegatively graded chain complexes, the sensible notion of $\mathrm{hom}\left(X,Y\right)$ is $ℤ$-graded! The negative grades creep in even if you hadn’t wanted them. This is how cochain complexes snuck into “week258”.

And of course homs are closely related to actions…

That was after he explained that $ℤ$-categories are all about stabilization.

take a 1-tuply monoidal 0-groupoid. That’s a group. It lives in degree 0.

The fact that it is 1-tuply monoidal means we can suspend it to a 1-groupoid with a single object.

Now the group itself is in degree 1. And the single object is in degree 0.

But let’s think about this: it is unfair to regard that single object just as a dot $•$.

Rather, we should think of the group as being the group of automorphisms of that single object. The shifting process in which we regard a monoidal $n$-thing as an $\left(n+1\right)$-thing here really makes manifest that a monoidal thing usually arises as the endomorphisms of some other thing.

A doubly monoidal 0-groupoid is an abelian group. We shift it up twice to obtain a 2-category with a single object and a single 1-morphism. And we should think of the abelian group really as being the automorphism group of that single morphism.

I am probably (hopyfully, even) stating a triviality in a verbose way. But it took me some thought process to arrive at this:

the game we are plaing here can be regarded as saying this:

if I have a $k$-tuply monoidal $n$-category, I can, instead of shifting it up to a $k+n$-category with single $p$-cells for $p, think of it as being something with cells in degree $0$ up to $n$, with something sitting in degree -1 to $-p$.

And moreover, as we have seen above, this stuff in the negative degrees is really nicely thought of as something that the stuff in the non-negative degrees acts on.

Now, as John explains, when both $k$ and $n$ go all the way through infinity, we arrive at $ℤ$-categories.

But if the above is right, it would mean that a good way to think of a stable $\infty$-groupoid is as the automorphisms of some infinity-thing hidded in the negative degrees.

Okay, that’s what i wanted to say. When the thought occurred to me I found it useful. Now that I have typed it into a comment box it feels a little pointless. But I’ll post it anyway.

Posted by: Urs Schreiber on December 5, 2007 10:37 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

I could try to claim that I’m starting to become an expert on things ${A}_{\infty }$, but given that Jim Stasheff is an avid commenter here, I don’t quite dare to. :)

However, I have read the Lu-Palmieri-Wu-Zhang [LPWZh] paper mentioned in the exposition backwards and forwards. On the face, what LPWZh try to do is to take the survey articles by Bernhard Keller, outlining the use of ${A}_{\infty }$-algebras in representation theory, and widening the scope of their proven usability while actually proving the many unproven and interesting statements that Keller makes.

At the core of this lies two different theorems. One is the Kadeishvili theorem (which in various guises has been proven by everyone involved with ${A}_{\infty }$-algebras, and a few more, in my impression ;) that says that you can carry ${A}_{\infty }$-algebras across taking homology. Kadeishvili’s argument specializes to the case where you start with an ${A}_{\infty }$-algebra with only ${m}_{1}$ and ${m}_{2}$ are non-trivial – i.e. a plain old dg-algebra. For higher generality, you’d probably want to turn to the Homology Perturbation Theory crowd with Stasheff, Gugenheim and Huebschmann among the more famous names…

Hence, since if we take graded endomorphism algebra of a resolution of $M$ and introduce the “homotopy differential”: $\partial f=df+fd$, then cycles are chain maps and the homology picks out exactly the algebra cohomology over the appropriate module category. Thus, we get Ext as the homology of a dg-algebra, and thus, Ext has an ${A}_{\infty }$-algebra structure.

The second cornerstone of these papers is the Keller higher multiplication theorem: if the ring $R$ is sufficiently nice, then the ${A}_{\infty }$-algebra structure on ${\mathrm{Ext}}_{R}^{*}\left(M,M\right)$ for some appropriate module $M$ will allow you to recover a presentation of $R$ explicitly.

I hope this answers your question about the origin of their ${A}_{\infty }$-algebra structure.

Posted by: Mikael Vejdemo Johansson on November 26, 2007 8:00 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Thanks, Mikael — that’s extremely helpful!

I think this means that the ${A}_{\infty }$-algebra structure that everyone uses on ${\mathrm{Ext}}_{R}^{*}\left(M,M\right)$ agrees with the ‘conceptually simple’ one that I proposed in “week258”.

My way — for which the details haven’t been worked out — was to associate an ${A}_{\infty }$-category to any sufficiently nice abelian category. For simplicity, let’s start with the category of $R$-modules for some ring $R$. Then my idea was to build an ${A}_{\infty }$-category with $R$-modules as objects, but a new hom, the ‘derived function complex’

$\mathrm{HOM}\left(x,y\right)=\mathrm{hom}\left(Px,y\right)$

Here $Px$ is a nice functorial projective resolution of $x$, say the free resolution. We think of $Px$ and $y$ as chain complexes of $R$-modules, and let $\mathrm{hom}\left(Px,y\right)$ be the cochain complex of abelian groups where the $n$-chains are degree-$\left(-n\right)$ maps of graded $R$-modules from $Px$ to $y$… not necessarily chain maps!

I believe there’s a reasonable way to define a composition

$\circ :\mathrm{HOM}\left(x,y\right)\otimes \mathrm{HOM}\left(y,z\right)\to \mathrm{HOM}\left(x,z\right)$

and I think this gives an ${A}_{\infty }$-category. It could in fact be a mere differential graded category, which would make what I’m saying sound a bit pompous. But either way, this would make

$\mathrm{HOM}\left(x,x\right)$

into an ${A}_{\infty }$-algebra (for example a dg algebra). Then, taking cohomology and using the Kadeishvili theorem, we see that

${\mathrm{Ext}}^{*}\left(x,x\right)={H}^{*}\left(\mathrm{HOM}\left(x,x\right)\right)$

is an ${A}_{\infty }$-algebra.

I guess the only difference between this stuff and what you’re saying is that I’m using

$\mathrm{hom}\left(Px,x\right)$

where you are using

$\mathrm{hom}\left(Px,Px\right)$

Hmm! These should be equivalent as ${A}_{\infty }$-algebras, but the multiplication is obviously associative ‘on the nose’ in yours, so yours is simpler.

$\mathrm{HOM}\left(x,y\right)=\mathrm{hom}\left(Px,y\right)$

maybe I should define

$\mathrm{HOM}\left(x,y\right)=\mathrm{hom}\left(Px,Py\right)$

to make composition easier to define, and get a dg category instead of an ${A}_{\infty }$-category. It’s probably ‘equivalent’ (in some relaxed sense of equivalence suited to ${A}_{\infty }$-categories), but more manageable.

Indeed, everything I’m doing has probably been done better already by the experts.

Posted by: John Baez on November 26, 2007 11:21 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

I’m not sure I understand precisely what you’re saying in this comment, but this is what I think you’re getting at. As you guessed, what you want to do is the dg-version of derived categories. (I’m not working on so much sleep, so don’t completely trust everything I’m about to say). Given an Abelian category, you can get a dg category by defining ${\mathrm{Hom}}^{k}\left(F,G\right)$ by the set of maps from $\left\{{f}_{i}:{F}^{i}\to {G}^{i+k}\right\}$. (Or maybe $-k$ – I can never remember.) The differential on this complex is $\left(df{\right)}_{i}=\delta {f}_{i}±{f}_{i+1}\delta$ Thus, closed maps are chain maps and exact maps are homotopic to zero. This is a nice dg-category.

The next thing you want to do is to “derive” this category, ie, invert the quasiisomorphisms in this setting. There are various ways to do this (cf, Keller or Drinfeld) – the most intuitive for me is that you want to kill the complexes that have no cohomology. Thus, (at least morally) you add a morphism to make the identity morphism of those complexes exact.

The resulting dg-category is a cooler version for the derived category. You can take cohomology and get an ${A}_{\infty }$ enhancement of the derived category.

For these ${A}_{\infty }$ algebras, life is much simpler. In particular, if you have a quiver algebra, there’s a nice distinguished set of simple representations associated to the nodes (if there are no directed loops, these are the only simples). These simples have a very nice projective resolution such that the ${\mathrm{Ext}}^{1}$s between them are the arrows in the quiver. What Keller claimed and was proven by Lu et al (and also by Ed Segal) is that you can get the whole quiver algebra out of the ${A}_{\infty }$ algebra of Exts of these simples. This algebra is the one coming from the dg-algebra that is the RHom complex (your HOM unless I misread).

A nice way to think about it is to take the dual of the cobar complex (really a slight modification taking into account the quiveriness) of this ${A}_{\infty }$ algebra. As stated below, this is a dg-algebra. Unless I’m misremembering, it’s zeroth cohomology is precisely the quiver algebra. (It’s always seemed to me that this story should reduce to some version of the idea that cobar bar is homotopic to the identity, but I’m not sure to what extent that’s true.)

Posted by: Aaron Bergman on November 27, 2007 2:48 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

My understanding (?) is that the derived cat is at the level of homotopy classes of the maps of resolutions - hence throwing away info again. The A_\infty cat structure using the resolutions themselves is where it’s at - or should be.

Which paper of Lu et al and of Ed Segal?

Cobar bar equiv identity needs a slight assumption I think, e.g. applied to connected things.

Bar cobar needs simply connectivity ???

Posted by: jim stasheff on November 27, 2007 2:52 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

The Lu et.al. paper is the A-infinity for Ext algebras preprint by Lu, Palmieri, Wu and Zhang. They work out proofs for most of the statements in Keller’s 2000 and 2001 papers (Introduction to A-infinity algebras and modules, and A-infinity algebras in representation theory – IIRC) for the case of an abelian group graded local ring, including the statement that the A-infinity structure maps ${m}_{n}{\mid }_{{\mathrm{Ext}}^{1}\left(k,k{\right)}^{\otimes n}}:{\mathrm{Ext}}^{1}\left(k,k{\right)}^{\otimes n}\to {\mathrm{Ext}}^{2}\left(k,k\right)$ are dual, in this setting, to the inclusion maps ${\mathrm{Ext}}^{2}\left(k,k\right)\to {T}^{*}{\mathrm{Ext}}^{1}\left(k,k\right)$ of relations into the tensor algebra over the generators of the ring $R$ we started with. For this to work, we need nice enough rings at the start - Keller’s paper show that quiver algebra quotients are good enough, and the meat of Lu-Palmieri-Wu-Zhang is in that these local rings also are good enough.

I have no idea what the Segal paper is, though.

Posted by: Mikael Vejdemo Johansson on November 27, 2007 4:24 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Aaron recalled how to define the Hom-complex ${\mathrm{Hom}}^{•}\left(F,G\right)$ of two chain complexes $F$ and $G$, namely ${\mathrm{Hom}}^{n}\left(F,G\right)={\oplus }_{k\in ℤ}\mathrm{hom}\left({F}_{k},{G}_{k-n}\right)$ with the differential being the graded commutator with the differentials on $F$ and $G$.

Last time I checked this I seemed to run into some sign issue when comparing this differential with the one on ${F}^{*}\otimes G$, which should coincide with the internal hom for the case that $F$ and $G$ are suitably of finite rank in each degree.

Maybe I was just being confused, but I could fix that sign only after introducing a sign in every second differential for ${F}^{*}$ and then looking at $\left(F\otimes {G}^{*}{\right)}^{*}$.

Did I just completely screw up or does anyone recognize this issue?

Posted by: Urs Schreiber on November 27, 2007 10:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Just apply the rule rigorously:
df = \pm fd depending on the degree fo f
and f \in graded hom has a degree

don’t feel bad! in the early days of the revival of foliation theory led by Bott and Haefliger, people were similarly puzled by incomaptible signs
which implies 1 = -1

Posted by: jim stasheff on November 28, 2007 2:03 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Just apply the rule rigorously:

I think I did just that. And still had a sign problem.

Okay, I am going to type that computation now in full detail. Either it comes out right this time and I am saved, or it does not, then I’ll ask you to have a look.

Posted by: Urs Schreiber on November 28, 2007 2:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Okay, I am going to type that computation now in full detail. Either it comes out right this time and I am saved, or it does not, then I’ll ask you to have a look.

Well, there is not much to compute. I must be being dense and missing something obvious. So this will be embarrassing. But I am eager to get this straightened out. So here is my reasoning:

This is also page 13 of this pdf, where you can check the conventions that I am using.

Okay, by the usual rule of web discussion, as soon as I will have posted this and made a fool of myself in public, I will see my mistake myself.

So let’s see…

Posted by: Urs Schreiber on November 28, 2007 3:47 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

You’re scaring the kids, Urs! (p.s. this reminds me of those weird drawings John Nash used to draw on the window of the library in A Beautiful Mind).

Posted by: Bruce Bartlett on November 28, 2007 5:32 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

You’re scaring the kids, Urs!

It’s maybe the kind of thing next to impossible to communicate efficiently by exachnging electronic text messages .

On a blackboard no kid would be scared by this.

Because the point I am puzzled about is so very elementary. It’s a pity that a glance at these equations doesn’t reveal that when looking at them in detail there is not much going on here.

So I should say it in words:

when we form the hom-complex, we insert a sign depending on the difference of the degrees of two parts of $V$ and $W$,

but when we tensor we insert a sign depending just on the degree of a part of $V$.

I am just trying to make somebody say: “yeah, Urs, but you dork forgot that in the $n$th line you pick up another sign due to xyz.”

Or something like that.

It’s still the issue I discussed with Todd Trimble here.

Posted by: Urs Schreiber on November 28, 2007 6:30 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

I have a feeling that the problem is that there are unacknowledged symmetry isomorphisms which haven’t been factored in.

The business of signs can be an exquisite torment, and the only cure which works for me is just to be relentlessly categorical in one’s approach. So, suppose that the cochain complex Hom is set up so that

$\mathrm{hom}\left(U\otimes V,W\right)\cong \mathrm{hom}\left(U,\mathrm{Hom}\left(V,W\right)\right)$

where $\mathrm{hom}$ denotes the “external hom” (valued in $\mathrm{Vect}$ or $A$-$\mathrm{Mod}$ or something) and $\mathrm{Hom}$ the “internal hom”. This says $\mathrm{Hom}\left(V,-\right)$ is right adjoint to $-\otimes V$. We get in particular an evaluation map

$\mathrm{Hom}\left(V,W\right)\otimes V\to W$

which, in order to preserve the differentials, must satisfy

$\left(df\right)\left(v\right)+\left(-1{\right)}^{p}f\left(dv\right)=d\left(f\left(v\right)\right)$

where $f$ and $v$ are homogeneous elements with $f$ of degree $p$. Hence we calculate

$df:v↦d\left(f\left(v\right)\right)-\left(-1{\right)}^{p}f\left(dv\right)$

which agrees with your formula, I guess.

The category of cochain complexes is symmetric monoidal closed (never mind this business of star-autonomy, for the moment). Define the dual ${V}^{*}$ to be $\mathrm{Hom}\left(V,I\right)$. Whether or not the category you’re interested in is compact closed, we always have, in an smc category, a canonical map

${V}^{*}\otimes W=\mathrm{Hom}\left(V,I\right)\otimes W\to \mathrm{Hom}\left(V,W\right),$

and this will be an isomorphism when the category is compact closed – but never mind that! We only care for now how this canonical map is actually defined and how it gets along with signs.

By the $\mathrm{Hom}$-$\otimes$ adjunction as stated above, this map corresponds to a map

$\mathrm{Hom}\left(V,I\right)\otimes W\otimes V\to W,$

and the map we are after here is actually a composite involving a symmetry isomorphism:

$\mathrm{Hom}\left(V,I\right)\otimes W\otimes V\stackrel{1\otimes \sigma }{\to }\mathrm{Hom}\left(V,I\right)\otimes V\otimes W\stackrel{\mathrm{eval}\otimes 1}{\to }W.$

I think the need to incorporate that extra symmetry isomorphism introduces an extra sign which should resolve the difficulty. Let’s both sit in a dark room for a while and work through it slowly and carefully and compare notes, but it should all work out fine, I think.

Whether this canonical map is an iso is of course a separate matter which devolves on finiteness assumptions, and isn’t really the issue here.

Posted by: Todd Trimble on November 28, 2007 8:38 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Thanks Todd, that helped.

I’ll go through my notes now and clean them up a bit using the hints you just provided.

After that it’ll be getting close to midnight here, and one shouldn’t try to catch a sign after midnight, but let’s see how it goes.

Thanks a lot. I appreciate it.

Posted by: Urs Schreiber on November 28, 2007 8:48 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Not wishing to belabor this, but there may be slight differences in sign conventions between us from the get-go. So what I was trying to suggest is more about means than ends: instead of starting with a whole slew of sign definitions, one for each component of structure of a symmetric monoidal closed category, I think it’s a good idea to start by using a more parsimonious approach, getting the signs nice and settled just for the monoidal part first, and then see how this forces all the rest to fall into place:

• First decide on what the sign convention should be for the differentials of tensor complexes of cochain complexes. My own preferred convention is to use $d\left(v\otimes w\right)=dv\otimes w+\left(-1{\right)}^{p}v\otimes dw$ where $v$ and $w$ are homogeneous and $v$ is of degree $p$. But you may be using a different convention.
• Using whatever convention you choose for the tensor product, and without peeking at anything else, figure out what this forces the differentials on internal Homs to look like (by chasing the diagram which says that $\mathrm{eval}:\mathrm{Hom}\left(V,W\right)\otimes V\to W$ respects differentials). For extra good measure, also chase the diagram which says that the symmmetry $\sigma :V\otimes W\to W\otimes V$ respects differentials. It should involve $\left(-1{\right)}^{pq}$ of course, but it doesn’t hurt to see how it’s forced from first principles.

If all works out as it should, there will be nothing tricky about associativities and unit isomorphisms, and you will have gotten all the sign stuff down pat for all of the closed symmetric structure, by construction.

Again, sorry if this is belaboring points well-known to you – it’s just me talking to myself about the need for a disciplined approach to these (sometimes really vexing!) issues.

Posted by: Todd Trimble on November 28, 2007 9:51 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Again, sorry if this is belaboring points well-known to you

Please don’t hesitate to belabor this to your heart’s content. It is precisely exchange like that which I was hoping for.

I do perfectly agree with everything you say, and if my approach was less systematic than yours then not because I intended it that way. I am all in favor of the most abstract and elegant way that you can think of for handling these things (I do will, however, want to and need to spell out all signs in full detail, too, in my notes, since parts of that is supposed to serve in a seminar we have).

As I predicted, I spent the end of this central-european day cleaning up my notes a bit based on your reactions, and that did involve changing one sign convention at one point, which seems to be the one (or at least one) you are referring to.

Then, as also predicted, I became too tired for chasing signs around and instead got carried away with writing up further stuff on the physics of BV.

It’s really kind of nice, you see: whatever that BV formalism is, those physicists keep talking about, the upshot is this nice general abstract nonsense statement:

BV formalism is what you get when forming the space of fields not as the external, but as the internal hom in cochain complexes.

I spent some time doodling around and creating this figure illustrating that point. Now I need to call it quits and catch some sleep.

I’ll get back to sorting out that sign issue tomorrow. Thanks very much for all your feedback.

Posted by: Urs Schreiber on November 28, 2007 11:47 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

v^* is not well defined
but V^* is okay but d^* involves a sign
but I haven’t stuck with this long enough to be sure that resolves your problem

anecdote: I sweated out the signs in my thesis at Hogwarts and then one of the examiners asked why I did; he said it was obvious that there existed a set of signs!!

Quillen also at times finesses the issue

Posted by: jim stasheff on November 29, 2007 1:18 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

In the chapter “Notes on Supersymmetry” in the book

• Quantum Fields and Strings: A Course for Mathematicians, Vol. 1. AMS and IAS, 1999

Deligne and John Morgan mildly remark, “Keeping track of signs can be a nuisance” (page 47). By the time you get to page 230 (at the end of Supersolutions), Deligne and Daniel Freed cry out in exasperation, “Writing this has been an absolute cauchemar des signes! Standard differential geometry has some bad signs, classical field theory has tricky sign conventions, and odd variables add a whole new level of complication.”

By the way, they advocate a categorical approach, in effect doing things abstractly so as to let the signs look after themselves, as far as possible.

Posted by: Todd Trimble on November 29, 2007 4:28 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Well, I oscillated somewhat between including it and not including it. (I started the discussion with noticing that we need it. :-) But including it, which I do above, does not solve the problem I have.

I was busy all day until now with other things. Will now try to continue thinking about straightening this out.

But by the way: why is it we are so sure that chain complexes are compact closed as opposed to just star-autonomous?

Maybe Todd already tried to tell me, but if so, I didn’t get it yet.

I assume there is some literature available where the statement is made and, if so, maybe even demonstrated?

Posted by: Urs Schreiber on November 29, 2007 5:00 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

But by the way: why is it we are so sure that chain complexes are compact closed as opposed to just star-autonomous?

Well, they’re neither. :-) That is, not without extra finiteness assumptions (which I’ll restate, since I’m not sure I made myself clear earlier).

• The category of chain complexes with the usual tensor product is symmetric monoidal closed. (This is the locus of all the sign conventions. The stuff below on compact closure is phrased in terms of additional conditions on symmetric monoidal closed categories, and by the phrasing requires no further elaboration on sign conventions.)
• By a star-autonomous category, we mean a symmetric monoidal closed category equipped with an object $D$ such that the natural “double-dual” map $A\to \mathrm{Hom}\left(\mathrm{Hom}\left(A,D\right),D\right)\right)$ corresponding under $\mathrm{Hom}$-$\otimes$ adjunction to the composite $A\otimes \mathrm{Hom}\left(A,D\right)\stackrel{\sigma }{\to }\mathrm{Hom}\left(A,D\right)\otimes A\stackrel{\mathrm{eval}}{\to }D$ is an isomorphism. In the case you are considering, the $D$ coincides with the monoidal unit (a 1-dimensional space concentrated in degree 0).

I will define a chain complex $\left\{{V}_{n}\right\}$ of vector spaces to be finite if each ${V}_{n}$ is finite-dimensional, and ${V}_{n}$ is 0 for all but finitely many $n$ (i.e., if the total space is finite-dimensional). Then the double dual map above is an isomorphism, and therefore finite chain complexes do form a star-autonomous category.

But moreover,

• A star-autonomous category is compact closed if $D$ is the monoidal unit $I$ (which we have in our case), and moreover the canonical maps (which exist in any symmetric monoidal closed category) $I\stackrel{\cong }{\to }\mathrm{Hom}\left(I,I\right)$ $\mathrm{Hom}\left(A,I\right)\otimes \mathrm{Hom}\left(B,I\right)\to \mathrm{Hom}\left(A\otimes B,I\right)$ endow $\mathrm{Hom}\left(-,I\right)$ with the structure of strong monoidal functor. (It’s already a lax monoidal functor, by a coherence theorem for symmetric monoidal closed categories. So the only issue here is whether the second map is invertible.)

It is not true that every star-autonomous category with $D=I$ is compact closed. (An example would be the category of ordinary sup-lattices [in $\mathrm{Set}$].) But the category of finite chain complexes of vector spaces is compact closed.

This comes down basic finite-dimensional linear algebra: it just boils down to the standard fact that the canonical map of vector spaces ${V}^{\star }\otimes {W}^{\star }\to \left(V\otimes W{\right)}^{\star }$ is injective (and hence an iso).

I haven’t said anything about $A$-modules (as opposed to vector spaces over a field) because I’m not quite sure what sorts of rings and modules you are considering. If the (commutative) ring has homological dimension 0 (where every module is projective), then you’re on safe ground.

I think you’re also on safe ground (i.e., all the isomorphisms needed to get compact closure obtain) if you’re considering modules over a cocommutative Hopf algebra. For you just need to check the isomorphisms at the vector space level.

Posted by: Todd Trimble on November 29, 2007 6:59 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Okay, thanks Todd.

extra finiteness assumptions

Yes. I also haven’t made myself clear before: I understand well that these finiteness conditions are a prerequisite. I wanted to understand this compact-closedness/star-autonomous business in the situation where everything is as tame as desired.

If the (commutative) ring has homological dimension 0 (where every module is projective), then you’re on safe ground.

Whatever my ring is, I want to talk here about the full subcategory, when it exists, of those chain complexes with the property that all of them are bounded and in each degree $\mathrm{hom}\left({V}^{k},{W}^{l}\right)\simeq \left({V}^{k}{\right)}^{*}{\otimes }_{A}{W}^{l}$.

So that was one problem in our conversation: I kept being worried about those signs while being deliberately non-worried about technical assumptions.

Okay, we have figured that out now. Thanks again.

Posted by: Urs Schreiber on November 29, 2007 7:26 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Let’s both sit in a dark room for a while and work through it slowly and carefully

I am beginning to feel really bad about this. If I am making a mistake, then at least I am making it pretty consistently.

I went again through my computations (the explicit scary way, sorry for that). And I do again find $\mathrm{hom}\left(V,W\right)=\left(V\otimes {W}^{*}{\right)}^{*}$ and $\ne {V}^{*}\otimes W$.

Could you maybe remind me why you think this cannot be correct?

Maybe if you could tell me which general abstract relation the above is violating, I could work out that general abstract relation in my scary way and this way discover what it is I am doing wrong.

Posted by: Urs Schreiber on November 29, 2007 6:17 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Well, I guess that’s the same as saying that it’s $W\otimes {V}^{*}$ instead of ${V}^{*}\otimes W$.

Er. Now I am pretty close maybe to both, having made a complete fool of myself and having actually solved my problem. :-)

Posted by: Urs Schreiber on November 29, 2007 6:29 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

All right, that’s it. I’ve got it.

My above, seemingly contradictory, explicit expression for ${V}^{*}\otimes W$ is actually correct. While it looks superficially different from $\mathrm{hom}\left(V,W\right)$, one can check that after applying the braiding isomorphism it is actually isomorphic to $\mathrm{hom}\left(V,W\right)$.

And in particular, $W\otimes {V}^{*}$ is in fact equal to $\mathrm{hom}\left(V,W\right)$. On the nose.

So the situation is this:

$\mathrm{hom}\left(V,W\right)=W\otimes {V}^{*}\simeq {V}^{*}\otimes W\phantom{\rule{thinmathspace}{0ex}},$

where by $\mathrm{hom}\left(V,W\right)$ I mean the specific one we have been talking about (as opposed to anything isomorphic to it) and the first equality sign is really meant to be an equality.

Phew, all right. Now on with the more substantial problems…

Posted by: Urs Schreiber on November 29, 2007 7:13 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

I haven’t been following the details of this battle, since it’s like one of those crazy shootouts at the end of a Hollywood action movie: bullets flying everywhere, lots of near-misses and sudden reversals of fortune… but you know the good guy will win in the end, so there’s no real suspense.

$1=1$ triumphs! $1=-1$ loses yet again!

I’m not sure which nasty supervillain Urs was fighting, but from his heroic words at the end of the battle I’m guessing it might be this:

If you apply a function $f$ to a vector $v$ and write the result as $f\left(v\right)$, you really should think of the evaluation map

$\mathrm{hom}\left(V,W\right)\otimes V\to W$

as

$W\otimes {V}^{*}\otimes V\to W$

not

${V}^{*}\otimes W\otimes V\to W$

Otherwise you need to switch ${V}^{*}$ and $W$, so you get punished with extra signs.

In short: if you write $f\left(v\right)$ for the result of applying $f$ to $v$, you really want to say

$\mathrm{hom}\left(V,W\right)=W\otimes {V}^{*}$

Taken by itself this equation looks a bit backwards: the $V$ and $W$ are getting switched. But this is a famous old problem — the bane of my existence! We read left to right, so we say “function from $V$ to $W$” and write “$\mathrm{hom}\left(V,W\right)$” for the space of those functions. But, we evaluate expressions like $g\left(f\left(v\right)\right)$ right to left! It’s a stupid combination of conventions.

If we wrote $\left(v\right)f$, the problem would go away. If we wrote $\mathrm{hom}\left(W,V\right)$ for functions from $V$ to $W$, the problem would also go away.

But, few people are willing to make either of these radical changes in notation. So, we’re stuck.

Unless we use string diagrams…

Posted by: John Baez on November 29, 2007 7:40 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Thanks, Wizard.

If there is any way this apprentice here could phrase his questions (like those about categories with negative morphisms or those about the subtleties of compact closure of chain complexes) in such a way that the Wizard replies to them before the apprentice embarks on a long adventurous quest, instead of right after he returns from such, I’d very much appreciate to know.

:-)

It was this kind of advice which I was seeking when I went back to Categorifying CCCs: Computation as a Process. There I stumbled across the notion of star-autonomy.

Even though I actually got it right early on, checking that $\mathrm{hom}\left(V,W\right)=\left(V\otimes {W}^{*}{\right)}^{*}=W\otimes {V}^{*}$ for chain complexes, I then got confused for stupid reasons.

Oh well.

Posted by: Urs Schreiber on November 29, 2007 8:09 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

A wizard’s words wouldn’t seem so wise were they to come before the apprentice engaged on a long, painful quest.

Seriously: I’d have helped sooner if I’d known what the trouble was. But it wasn’t clear until near the very end that the issue was this:

“Is it better to think of $\mathrm{hom}\left(V,W\right)$ as ${V}^{*}\otimes W$ or $W\otimes {V}^{*}$?”

If you’d been able to ask just that question, I could have answered it. But as usual in math, the key issue is surrounded by distracting fluff, and only becomes visible near the end.

If it makes you feel better, I’ve struggled with this issue many times! It came up while teaching my class on string diagrams for monoidal closed categories. For example, see the picture of the evaluation map on page 5 of the week 4 notes: I draw $\mathrm{hom}\left(A,Y\right)$ as a “ribbon” that resembles the string diagram for $Y\otimes {A}^{*}$.

Working in a closed monoidal but not braided or symmetric category is a very illuminating exercise, since you just don’t have the braiding

$A\otimes B\cong B\otimes A.$

A lot of things become harder, or impossible — but the things you can do, can often be done in just one way! This is then the right way, even in the braided or symmetric context.

It’s a bit like taking a visit to Edwin Abbbott’s Lineland, where people can’t switch places. It’s very claustrophobic at first. But you soon get used to it, and then the freedom of an extra dimension — which allows for a braiding — seems like a expensive luxury, to be used only when absolutely necessary.

Posted by: John Baez on November 30, 2007 7:34 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Hi, John.

just a few remarks about your stuff on Deligne’s conjecture. Unfortunately, technical details are important in this business.

First, we have to be careful about tensor product of operads. A very long standing question is. Let A be a E1-operad and B be a cofibrant E1-operad. Is it true that A\box B is an E2-operad? The answer is unknown, even though Dunn’s argument is correct and the tensor product of two little 1-cube operads is equivalent to the little 2-cube operad. Unfortunately, the theorem from Hu, Kriz and Voronov is based implicitly on an affirmative answer to the above question.

I think the history of Deligne’s conjecture is quite remarkable and complicated and still developing. The most conceptual and correct proof I know is provided by Tamarkin in

What do DG categories form?, Dmitry Tamarkin, math.CT/0606553

And it uses my up to homotopy Eckmann–Hilton argument. This argument is based on a techniques of compactification of configuration spaces and first was proposed by Getzler and Jones. I think I already wrote about it in a post to n-category cafe where Dolgushev’s work was discussed. Here is the reference to my lecture about Deligne’s conjecture:

http://www.math.mq.edu.au/~street/BataninMPW.pdf

Concerning your idea to construct an A-category using Hom(PX,Y), where PX is a projective resolution: it’s been done by me many years ago and in a more general situation. It is long story to tell but more or less I prove that your Hom functor is equivalent as a simplicially coherent bimodule to the homotopy coherent left Kan extension of the inclusion functor

Projective bounded chain complex → Bounded chain complex

along itself. Then the Kleisli category of this distributor has a canonical A-structure and this Kleisli category is equivalent in an appropriate sense to your ‘puffed’ category. In fact, the situation I consider in my paper is much more general and includes simplicial Quillen categories as a very special example. The paper is:

“Categorical strong shape theory” , Cahiers de Topologie et Geom. Diff., V.XXXVIII-1 (1997), 3-67.

and its companion

“Homotopy coherent category theory and A structures in monoidal categories, JPAA, 123 (1998), 67-103.

regards.
Michael.

Posted by: Michael on November 27, 2007 6:57 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Michael’s version is much more up to date and advanced than mine, still you young’uns might like to know some history.

Posted by: jim stasheff on November 27, 2007 3:03 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Thanks for the info, Michael!

(For those not in the know: this is Michael Batanin of $n$-category fame.)

I was very pleased to have understood the basic intuitive ideas underlying Deligne’s conjecture — they’d never been clear to me before. You’d told me about the difficulty of putting a cellular structure on the little $k$-cubes operad, and Tamarkin’s discovery of a mistake in the paper by Getzler and Jones. But, I didn’t know that was related to this:

A very long standing question is: let A be a ${E}_{1}$-operad and B be a cofibrant ${E}_{1}$-operad. Is it true that $A\mathrm{box}B$ is an ${E}_{2}$-operad? The answer is unknown…

Thanks for all the references. I hope I have the energy to learn more…

Posted by: John Baez on November 29, 2007 7:11 AM | Permalink | Reply to this

Some time ago Michael Batanin remarked:

A very long standing question is. Let $A$ be a ${E}_{1}$-operad and $B$ be a cofibrant ${E}_{1}$-operad. Is it true that $A\square B$ is an ${E}_{2}$-operad?

It seems that Lurie now claims to have a proof of this. See Additivity theorem.

Posted by: Urs Schreiber on December 17, 2009 2:23 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Thanks for the praise John. This is very close to my thesis subject, and I spent a large part of this month writing precisely the bits and pieces that introduce ${A}_{\infty }$ to non-experts.

One reason I talk about $\mathrm{hom}\left(\mathrm{Px},\mathrm{Px}\right)$ is precisely that then the chain composition is trivial and that with the differential I choose, the generic “higher operations are homotopies” become really obvious; we really do choose the higher operations when calculating such that they are homotopies between associators, and computing Ext as homotopy classes of chain maps makes this both more explicit and more tangible.

Posted by: Mikael Vejdemo Johansson on November 27, 2007 12:43 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

If you think of x as a deformation retract of Px, then either of your definitions of
HOM(x,y) should be homotopy equivalent to
hom(x,y). That said, we are back to the cat version of Kadeishvili (his was for algebras) or one of the original A_\infty
theorems: If A is an associative space or dg algebra and homtopy equivalent to Y just as a chain complex, then the algebra structure on A can be transferred to an A_\infty structure on Y (cf my birth certificate or most likely in MSS - my copy not handy.)

Posted by: jim stasheff on November 27, 2007 2:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Which Keller articles? Would one of the moderators convert Mika’s answer to links?

Posted by: jim stasheff on November 27, 2007 2:38 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

The Keller theorem is scattered (but never proven to my knowledge) in various articles available on his webpage. In particular, the ones with “representation theory” in the title have a statement, I believe. As for the other reference, the Keller paper on deriving dg categories is also on that page. The Lu et al paper is here and the Segal paper is here.

Posted by: Aaron Bergman on November 27, 2007 2:59 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

I wouldn’t call it scattered, but rather repeatedly stated. I know it occurs in both the HHA paper (also arXiv:math/9910179) and the ICRA Paper.

I am slightly surprised that you don’t seem to know about these, Jim - this is where I learned about ${A}_{\infty }$ to begin with, and they’ve been constant references for my own learning and work.

Posted by: Mikael Vejdemo Johansson on November 27, 2007 4:39 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Finally a question more to the main point of the above entry:

One way to say ${A}_{\infty }$-algebra is to say:

we have a chain comlex $A$ (in non-negative degree) together with a list of maps $\begin{array}{rl}{d}_{1}& :{A}^{*}\to {A}^{*}\left[-1\right]\\ {d}_{2}& :{A}^{*}\to \left({A}^{*}\otimes {A}^{*}\right)\left[-1\right]\\ {d}_{3}& :{A}^{*}\to \left({A}^{*}\otimes {A}^{*}\otimes A*\right)\left[-1\right]\end{array}$ etc. such that these things combined extended to $\Lambda {A}^{*}:={\oplus }_{k\in ℕ}\left({A}^{*}{\right)}^{\otimes k}$ define the structure of cochain complex on $\Lambda {A}^{*}$.

Somehow it feels like there must be a nicer way to say this, but without mentioning the word operad.

Posted by: Urs Schreiber on November 26, 2007 9:29 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Urs wrote:

Somehow it feels like there must be a nicer way to say this, but without mentioning the word operad.

What, you don’t like the word ‘operad’?

(Whoops, I just mentioned it!)

In week239 I said it sort of like this:

Making a graded vector space $L$ into an ${A}^{\infty }$-algebra is the same as making the free graded associative algebra on $\Sigma {L}^{*}$ into a differential graded associative algebra.

Here ${L}^{*}$ is the dual of the graded vector space $L$, and $\Sigma {L}^{*}$ is its ‘shifted’ or ‘suspended’ version.

(In the $ℤ/2$-graded context, Manin calls $\Sigma {L}^{*}$ the superdual of $L$.)

But you probably know all this, so probably this isn’t what you meant by ‘nicer way to say this’. Maybe you wanted a deeper reformulation.

Posted by: John Baez on November 26, 2007 11:36 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Urs - please use T for the tensor space, not Λ. Your di are quite clear; it’s the relations are nicely captured by D2 =0. What could be nicer than that? Of course all this was invented/discovered before we had operads! They were useful when the structures became more complicated.

Posted by: jim stasheff on November 27, 2007 3:07 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

First here is a great introduction to A-infinity categories for anyone like myself who needs clever pictures to keep track of the axioms:

Introduction to A-infinity algebras and modules

Second, I can’t help but notice a mysterious convergence of ideas in John’s excellent column this week. Searching for polycyclic aromatic hydrocarbon in the online encyclopedia of integer sequences leads of course to the hexagonal polyominoes. The restricted version of these gives rise to the sequence A002212, which is calculated by taking the binomial transform of the Catalan numbers. Of course the latter enumerate the vertices of the associahedra– which leads one to wonder whether the structure of those polytopes has anything to say about the organization of hydrocarbons.

Posted by: Stefan on November 26, 2007 10:02 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

That’s cool! The ‘mysterious convergence’ you mention is probably a result of my personality. I included those pictures of polycyclic aromatic hydrocarbons because I like beautiful complicated things made of simple building blocks… and this is also why I like ${A}_{\infty }$-algebras.

But, I didn’t suspect they were related, and I have no idea what to do with the relationship.

For now, all I can say is this:

I want somebody to invent a mathematical construction using the complex numbers (C), the quaternions (H), and the octonions (O), such that for any diagram of a carbohydrate I get some interesting mathematical object. I think this idea would be really sweet.

To do this, we need to figure out some sense in which C has four ‘bonds’, O has two bonds, and H has one. Then we can try combining (tensoring?) these algebras in a way that uses these bonds.

The powers of two are promising, but I always get stuck at this point. If I were John Conway I would dream up something clever and write a book based on this idea! But I’m not as smart as him, so I guess I’ll give up and throw out this idea to anyone who can do something with it.

Various chemical reactions should correspond to morphisms of some sort, so we can enjoy pondering things like

CH4 + O2 + O2 → CO2 + H2O + H2O

Posted by: John Baez on November 26, 2007 10:26 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Hmm. As complex vector spaces we have $C\cong {C}^{1}$, $H\cong {C}^{2}$, and $O\cong {C}^{4}$. So, it would work better mathematically if we took carbon to have one bond, hydrogen to have two, and oxygen to have four.

We can then define a ‘bond’ to be a specific sort of representation of the algebra of complex numbers on the vector spaces $C$, $H$, or $O$. The complex numbers $C$ has one obvious bond, since $C$ acts by multiplication on itself. $H$ gets two if we pick a vector space isomorphism $H\cong {C}^{2}$. $O$ gets four if we pick a vector space isomorphism $O\cong {C}^{4}$.

This lets us can ‘bond together’ some copies of $C,H$ and $O$ by treating them as modules of the algebra $C$, or the algebra ${C}^{2}$, or the algebra ${C}^{4}$, and then tensoring them.

Do you get the rules? For example, the diatomic molecule

$C-C$

is a notation for

$C{\otimes }_{C}C$

But hey! This is just C again. So, it ain’t like normal chemistry.

I leave as puzzle to work out this diatomic molecule:

H=H$\text{Unknown character}/p\text{Unknown character}\text{Unknown character}p\text{Unknown character}\mathrm{which}\mathrm{is}\mathrm{short}\mathrm{for}\text{Unknown character}/p\text{Unknown character}\text{Unknown character}p\text{Unknown character}$$H{\otimes }_{{C}^{2}}H$$\text{Unknown character}/p\text{Unknown character}\text{Unknown character}p\text{Unknown character}\left(\mathrm{By}\mathrm{the}\mathrm{way},I\mathrm{really}\mathrm{hate}\mathrm{how}\mathrm{this}\mathrm{implementation}\mathrm{of}\mathrm{TeX}\mathrm{has}\mathrm{symbols}\mathrm{for}$\mathbb{C}$\mathrm{and}$\mathbb{H}\$ but not the corresponding thing for the octonions. Can’t they afford a full set of blackboard-bold capital letters? Or it is just an evil plot to suppress octonionic mathematics?)

Posted by: John Baez on November 27, 2007 2:34 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

To some extent, biology, bioengineering and chemical physics via Faulon,Visco and Roe, Enumerating Molecules have started to do this on page 66 figure 17.

Other examples such “counting” can be found on a Google search of helicenic hydrocarbons polyhexes.

Posted by: Doug on November 27, 2007 3:38 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Perhaps we need some H2OCH as reagent. :-)

Posted by: jim stasheff on November 27, 2007 3:28 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

I’m not encouraged by the fact this idea depends on the accidental collisions of the first letters of three pairs of English words.

This is not usually a promising recipe.

Posted by: James Cranch on November 28, 2007 5:18 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

You seem to know the origins of the formula better than I

Posted by: jim stasheff on November 28, 2007 8:18 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

This is not usually a promising recipe.

I’ve seen mathematicians (Louis Kauffman, notably) start from less-sensible premises and manage to come up with something. Don’t count a free-association out too early.

Posted by: John Armstrong on November 28, 2007 9:50 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

James Cranch wrote:

I’m not encouraged by the fact this idea depends on the accidental collisions of the first letters of three pairs of English words.

This is not usually a promising recipe.

Right. But it’s not a crackpot theory of chemistry. It’s supposed to be an elaborate sort of joke: a correspondence between carbohydrates and some sort of mathematical structures built using normed division algebras. But, it’ll only be funny if works well.

Posted by: John Baez on November 29, 2007 2:51 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Speaking of Conway and chemistry and weird coincidences: I imagine many of you will have heard of Conway’s “look-and-say” sequence

1

11

21

1211

111221

and its weird “audioactive chemistry”. It turns out that every list in this sequence is a compound of atoms, which can be discovered by studying how they “decay”. In turns out there are exactly 92 such atomic elements, the same as the number of chemical elements in nature! (Edit: there are also “transuranic elements”, but asymptotically the probability of their occurrence is 0.)

To generate them, you start with the list 3 (“uranium: U92”), which ‘decays’ to 13 (“protactinium: Pa91”), which decays to 1113 (“thorium: Th90”); after a while you get cases where an element will decay to two or more elements. The only stable element is 22 (“hydrogen”), which occurs after 91 evolutions starting with uranium. The audioactive table can be found here, with preliminary discussion and chemical analysis here.

Also well-known is Conway’s Cosmological Theorem, which shows that the growth rate of the look-and-say sequence is asymptotic to ${\alpha }^{n}$, where $\alpha$ is an algebraic number of degree 71. You can get more information (including Sloane numbers) here.

Posted by: Todd Trimble on November 29, 2007 5:24 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Wow! See: this is the kind of crazy stuff Conway can pull off. When I try, it never works.

Posted by: John Baez on November 29, 2007 7:42 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

In re: two monoids on the same set:
It’s clear in John’s proof but it needs emphasizing: two monoids WITH THE SAME 1 on the same set

On my generals exam at PU, Fox asked why the fundamental group of a topological group was abelian. My response: because it’s a group for 2 different reasons. He was not amused.

Posted by: jim stasheff on November 27, 2007 3:37 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

If I’m not mistaken, it’s automatic that they have the same 1.

Theorem Let $S$ be a set. Let $\cdot$ be a binary operation on $S$ with two-sided identity $1$. Let $\star$ be a binary operation on $S$ with two-sided identity $e$. Suppose that $\left(a\cdot b\right)\star \left(a\prime \cdot b\prime \right)=\left(a\star a\prime \right)\cdot \left(b\star b\prime \right)$ for all $a,a\prime ,b,b\prime \in S$. Then $\cdot =\star$, $1=e$, and $\left(S,\cdot ,1\right)$ is a commutative monoid.

Proof Most of this is very well-known. The bit at issue here is proving that $1=e$; we have $1=1\cdot 1=\left(e\star 1\right)\cdot \left(1\star e\right)=\left(e\cdot 1\right)\star \left(1\cdot e\right)=e\star e=e.$ It’s also not very well-known that associativity comes for free, but I’ll leave that part as an exercise.

Posted by: Tom Leinster on November 28, 2007 12:52 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Touche’! I was thinking of the analog for homotopy commutative.

Posted by: jim stasheff on November 28, 2007 2:00 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

I forget if it was you or Eugenia Cheng who first pointed this out to me: two monoid structures on the same set satisfying the interchange law

$\left({a}_{1}\cdot {b}_{1}\right)\star \left({a}_{2}\cdot {b}_{2}\right)=\left({a}_{1}\star {a}_{2}\right)\cdot \left({b}_{1}\star {b}_{2}\right)$

automatically share the same unit. That’s why I omitted this hypothesis from my discussion of the Eckmann–Hilton theorem. You give the proof but not the easy way to remember it: arrange for a battle between the two units in such a way that both of them must win, and conclude they must be the same!

It resembles the usual proof that a monoid can’t have two different units:

$1=1\cdot e=e$

And if even that’s not enough to remember how to do this sort of proof, here’s a story that should do the job:

Two guys walk into a saloon. They both clam to be the fastest gun in the West! To settle the matter, you send them outside and arrange for them to fight a duel. Bang! Bang! It turns out they were both right. How could it be? They were both the same guy! It turns out you were seeing double. Hearing double, too! Too much whisky, I guess.

I didn’t know associativity came for free, but the proof is obvious now that you mention it. Very nice.

Posted by: John Baez on November 28, 2007 6:09 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Two monoids walk into the n-Category Cafe.

The first one is whistling the theme from “The Good, the Bad, and the Ugly.” The second one cites:

James T. Sandefur, The Gunfight at the OK Corral, Mathematics Magazine, Vol. 62, No. 2 (Apr., 1989), pp. 119-124

The first one counter-cites:

Amanda G. Turner, Convergence of Markov processes near saddle fixed points, Annals of Probability 2007, Vol. 35, No. 3, 1141-1171 and mentions that this considers sequences of Markov processes in two dimensions whose fluid limit is a stable solution of an ordinary differential equation. Here the processes are indexed so that the variance of the fluctuations of X_t^N is inversely proportional to N. The simplest example arises from the OK Corral gunfight model which was formulated by Williams and McIlroy [Bull. London Math. Soc. 30 (1998) 166–170] and studied by Kingman [Bull. London Math. Soc. 31 (1999) 601–606]. These processes exhibit their most interesting behavior at times of order log N so it is necessary to establish a fluid limit that is valid for large times.

The bartender says, in a Jamaican accent: “Calm down, Mon. Didn’t you know that Galois was shot to death in his intestine in a gun fight?” He then asks if the prior results can be categorified. He pours the two drinks on the house, from a Klein bottle.

The two monoids gulp down their drinks. The first says: “did you ever have the impression that you were in a humorous mathematical anecdote?”

The second squints his eyes, and lights a cigar. “Don’t get all recursive on me.”

The first says: “But that’s where it starts to get interesting.”

Suddenly, arrows are flying through the Western air…

My wife has been encouraging me to write a story called “Great Mathematicians of the Wild West.” But Pynchon’s “Against the Day does this better than I could. Has anyone here read it, and cracked up over the battles between the 4 Vectorists and the Quaternionists, or the song lyrics about mathematicians, such as the one that begins:

Her idea of banter
likely isn’t Cantor,
Nor is she apt to mumur low
Axioms of Zermelo,
She’s been kissed by geniuses,
Amateur Frobeniuses,
One by one in swank array,
Bright as any Poincare’,
And… though she
May not care for Cauchy,
Any more than Riemann,
We’ll just have to dream on…

Posted by: Jonathan Vos Post on November 28, 2007 10:22 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

(First time poster, six months lurker)

Jonathan -
Well there’s a funny thing. I’m 800 pages into Pynchon’s “Against the Day” and was just thinking of posting the same question.

Maybe it’s because I started reading it around about the same time that I discovered “This Weeks Finds”, but reading of Kit and Yashmeen’s adventures in Europe in the book remind me of John Baez’s recent travels there.

John - I really enjoyed your Octonions paper by the way.

Posted by: Colin Backhurst on December 2, 2007 11:52 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Thanks!

I haven’t read any Pynchon apart from Gravity’s Rainbow.

Posted by: John Baez on December 3, 2007 7:02 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Well its a tale of mathematicians, anarchists and spies, interspersed with the usual Pynchon comedy moments and songs.

Its similar to Gravity’s Rainbow, but its set in the political turmoil of the years at the end of the 19th Century leading up to the First World War.

It contains quite a lot of the changing maths and physics of the time (Quaternions versus Vectors, Aether, Relativity, hints of Quantum Mechanics etc). There’s a lot there.

Its hard to summarize it really, but if I had to put it in a phrase its about “Maths and politics”. So you might like it.

Posted by: Colin Backhurst on December 6, 2007 12:18 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

mathematicians, both real (e.g. Leibniz and Newton)
and fictional appear sporadically through Stepehnson’s baroque trilogy
esp in Con-fusion

Posted by: jim stasheff on December 6, 2007 3:15 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

Now that monoids have officially been anthopomorphised on this blog, I can’t resist pointing out that the Monoids starred in an old episode of Doctor Who! They were the treacherous servants to the last survivors of the human race, who flee the Earth 10 million years in the future as its orbits decays into the Sun. Have a look at the Wikipedia article here.

Wonderfully, the Monoids only have one eye — no doubt representing their unit! (No mention was made of their associativity.)

I have always thought that mathematical entities would make great characters in science fiction. As words, they are often appropriately grand and foreboding — just off the top of my head, the Magmas, the Equalisers and the Adjoint Functors would make great names for a race of marauding aliens! I wonder what our resident professional science fiction author thinks…

Posted by: Jamie Vicary on November 29, 2007 6:06 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

And, in a stunning coincidence, “The Magmas, The Equalisers and The Adjoint Functors” is a simply excellent band name.

Of course the name seems to indicate that it’s composed of members from other bands, which would make it a supergroup. :D

Posted by: John Armstrong on November 29, 2007 11:21 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

I think Magma and Equalizer would make great names for American Gladiators. (“Blaze! Nitro! Storm! Magma! And, The Equalizer!”)

I like the Adjoint Functors for a musical group – I’m imagining something like The Swingles, or The Mamas and the Papas.

Posted by: Todd Trimble on November 30, 2007 12:07 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

I can hear the commentators now: “The Equalizer is taking his opponent to the limit!”

Posted by: John Armstrong on November 30, 2007 3:25 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 258)

When I was an undergrad at Princeton, some math grad students formed an impromptu but very distinctive rock band called Monoid. Why? Because it was a ‘semigroup with identity’.

Posted by: John Baez on November 30, 2007 7:42 PM | Permalink | Reply to this
Read the post On BV Quantization, Part VIII
Weblog: The n-Category Café
Excerpt: Towards understading BV by computing the charged n-particle internal to Z-categories, secretly following AKSZ.
Tracked: November 30, 2007 10:32 AM

Re: This Week’s Finds in Mathematical Physics (Week 258)

In week258 jb said:

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 cochain complex whose cohomology is familiar: it’s the “Hochschild cohomology” of A.

It’s interesting, becuase I remember learning about this stuff in week209. However, when I looked there just now I discovered that it contained nothing at all ‘derived’. I guess that week209 must have laid some groundwork so that when I then read Sections 2.4 and 3.2 of Ganter and Kapranov’s paper it all seemed more obvious (actually I think it was tied in with lots and lots of derived things I was thinking about at the time). It’s strange how it’s difficult to remember exactly how you learnt things…

Posted by: Simon Willerton on December 4, 2007 11:38 AM | Permalink | Reply to this

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