The n-Category Café

Thanks David!

I fixed the link. Thanks for catching that.

Looking in our existing math toolkit for tools to attack problems in biology may be missing the boat.

After all, calculus etc were invented to attack problems in physics.

Sturmfels’ paper asks not “what mathematics can do for biology” but “what biology can do for mathematics”. This way of posing the question tends to let mathematicians continue thinking about concepts they already like… and find connections to biology here and there. But it’s possible that ultimately biology will do more for mathematics if mathematicians take a more “submissive” role and try very hard to understand biology. It may not work, but it may lead to something truly new. Something like this is what led mathematicians thinking about physics to invent calculus. Newton’s Principia did contain new theorems about Euclidean geometry, but that’s not what we remember it for.

Thinking about this gave me new respect for Gauss’ line about mathematics being the “queen and handmaiden of the sciences”. The submissive role of handmaiden may be inseparable from the majestic role of queen.

Thanks for pointing out the paper A symbolic computational approach to a problem involving multivariate Poisson distributions.

It’s about situations where you have a bunch of independent Poisson random variables $X_j$ with known Poisson parameters, and you want to know the distribution of one conditioned on the values of some linear combinations of these variables.

It give some ways to efficiently compute this distribution using ‘Wilf-Zeilberger theory’ and the ‘Apagodu-Zeilberger algorithm’. The authors wrote a Maple package that does the job.

Since I don’t know Wilf-Zeilberger theory or the Apagodu-Zeilberger algorithm, it’s hard for me to get deeply interested in this material.

However, Section 6, on biochemical applications, is fascinating to me! It’s about ‘chemical reaction networks’. Over on Azimuth, I’m just about to start explaining the connection between chemical reaction networks and quantum field theory. So, this is a topic I’m eager to hear more about.

The connection is that chemical reaction networks satisfying a certain technical condition have stable equilibrium states where the number $X_j$ of molecules of the $j$th type is a Poisson random variable, and these random variables are independent. So, the work in this paper applies.

But what I really like is that Section 6 is a self-contained detailed introduction to chemical reaction networks, with lots of examples, and references to a bunch of the most important papers on the subject!

In case I was a bit vague above, my point was that other than a couple of theorems that justify some problem transformations, it’s basically about symbolic algorithmic techniques that produce some recurrences and initial conditions for any given system. They look sufficiently complicated that I think most mathematicians would abandon this approach if forced to do it by hand. It’s clearly rigorous and useful if one is interested in results for a given network, but is that kind mathematics the kind of thing that will have an effect within “mathematics for it’s own intrinsic interest” (I don’t want to say pure maths, because that’s not quite what I mean)?

Dave wrote:

They look sufficiently complicated that I think most mathematicians would abandon this approach if forced to do it by hand. It’s clearly rigorous and useful if one is interested in results for a given network, but is the kind of thing that will have an effect within “mathematics for its own intrinsic interest” […]?

Maybe you know this already, but:

Doron Zeilberger, one of the coauthors of that paper, is famous for using computers to prove complicated identities involving hypergeometric series. He credits his computer as a coauthor on many of his papers under the name Shalosh B. Ekhad. He argues that programming is even more fun than proving, and more importantly it gives as much, if not more, insight and understanding. He also argues that meta-understanding will replace understanding:

Is is high time that we, mathematicians, give up that obsession to follow all the details. Mathematics, even before computers, got so complicated, that it is hopeless to try and “understand” (in the local sense) all the results that one uses. It is fair game to use a proved theorem, approved by experts, as a macro (or “black box”), as long as one understands the statement, and the conditions under which it holds, without feeling obligated to (locally) understand its proof.

So even with human-generated mathematics, it is no longer possible to have perfect “local” “understanding. With computer-generated mathematics, it is entirely hopeless. We should learn to trust the output of the computer as a “black box”. Of course, computers, at present, are still programmed by those feeble-minded creatures called humans, so they should not be blindly trusted. But with the right “architecture”, both for hardware and software, and lots of quality-control tests, we should learn to trust computers more and more.

We also have to change our notion of “beauty”. Dear Uncle Paul Erdös, I have bad news for you. Most theorems do not have proofs in your divine book, even if their statements are “beautiful” (e.g. the Four Color Theorem). You probably already knew it, since I am sure that Kurt Gödel told you that there exist short (i.e. “pretty” in your eyes) statements with arbitrarily long proofs (i.e. “ugly” to your eyes), but I bet that you dismissed it as “metamathematical nonsense”. So indeed, if the definition of a “beautiful” proof is a “proof from the book”, i.e. that you can read and understand all the details of in five minutes, then, luckily, most results are not pretty, since, if you, a mere human, can understand it so well, it can’t be very deep.

[You may retort that an easy-to-follow proof is not necessarily easy-to-find, and you would be right, but if its proof is short, it is, at any rate, a posteriori trivial, even if it not a priori so.]

But, we humans should not despair. We just have to adjust, and learn to live with, computer-generated mathematics, and tweak both our notions of “understanding” and “beauty”. As for understanding, we should give up on “local”, micro-managing, understanding, and be happy with the global kind, i.e. trade understanding with meta-understanding, and if necessary, even with meta-meta-understanding. Understanding the algorithm that generated the proof, or even merely understanding the meta-algorithm that generated the algorithm that generated the proof.

So, it’s possible that what mathematicians will find most interesting about this paper is whether it advances Zeilberger’s controversial vision of mathematics.

I don’t think we can draw any general conclusions about math inspired by biology or chemistry from this case. The only real link between this paper and those subjects is the ‘chemical master equation’. And as I hope to show soon on Azimuth, this equation can also serve as the basis for mathematical reflections much more congenial to lovers of elegance as traditionally defined: we’ll see how it’s related to quantum field theory, category theory, algebraic geometry, and the like.

I might be inadvertantly coming across as sceptical in these comments, whereas I’m really trying to think about what it would look like for a different field (say biology) to lead to new mathematics. As mentioned, one of the interesting thing about this paper is the way that, despite being perfectly precise and rigorous, it doesn’t look like your typical mathematics paper (threre’s no new organising concepts, no significant theorems/classifications, etc). It would be interesting to look back and see what the assimilation-into-general-mathematics process for papers similar to this will turn out to be in the coming years.

(Personally, I’m both fascinated and inspired by Zeilberger and collaborators stuff, but my interests are influenced as much by interest in computational issues as mainstream mathematical views.)

My enquiries on the category mailing list led back to Lawvere’s thesis as the origin of the basic idea. Apart from that, whilst there was a fair bit of folkloric stuff, there were very few actual references. Indeed, in the published version of Comparative Smootheology then I simply refer to Lawvere’s thesis.

The stuff on the nLab page on Isbell envelope was mostly me working out the details for Froelicher spaces (see the nLab page on http://ncatlab.org/nlab/show/Fr%C3%B6licher+spaces+and+Isbell+envelopes) but in a more general context. I make no claim to originality, but I also did not have any particular sources for that.

The only other reference that I can think of that might be relevant is a paper by Isbell called Adequate subcategories (details on lSpace: http://ncatlab.org/lspace/show/Adequate+subcategories). But it’s been a while since I looked at that paper so I’m not sure how relevant it will be.

I’ve not been following this discussion so I’m afraid I don’t have anything more to contribute. Is there an easy way to get up to speed? The preceding comments are full of stuff I don’t understand, and I’m unsure of the context! What little I did follow sounded interesting, and when I was looking at Isbell envelopes then I found it an attractive idea so I’d like to see if it is relevant or not.

Okay, so I’ve now figured out how the thing Tom called $I(A)$ is the same as thing that Andrew called the “full subcategory of objects satisfying Isbell duality in the Isbell envelope”. In case anyone’s interested, I’ll explain it here. I hope to say what this has to do with metric spaces in another post. The relation between the two comes down, essentially, to the following category-theoretic observation – it took me a long time to spot this and is now kind of obvious to me – it’s probably some standard thing.

Theorem If $L: C\to D$ and $R: D\to C$ form an adjunction $L\dashv R$ then the following three categories are equivalent:

• the full subcategory of $C$, denoted $Fix(R L)$, consisting of the objects $P\in C$ such that the unit map $\eta_P\colon P\to R L(P)$ is an isomorphism;

• the full subcategory of $D$, denoted $Fix(L R)$, consisting of the objects $Q\in D$ such that the counit map $\epsilon_Q\colon L R(Q)\to Q$ is an isomorphism;

• the category, denoted $Dual(L,R)$, where the objects of $Dual(L,R)$ consist of a pair $P\in C$ and $Q\in D$ together a pair of isomorphisms $\alpha\colon P\stackrel\sim\to R (Q)$ and $\beta\colon L (P)\stackrel\sim\to Q$ which are adjoint, and where the morphisms of $Dual(L,R)$ from $(P,Q,\alpha,\beta)$ to $(P',Q',\alpha',\beta')$ are the obvious things [oh, well if you insist, they’re given by a pair of morphisms $f\colon P\to P'$ and $g\colon Q\to Q'$ such that $R(g)=\alpha' f \alpha^{-1}$ and $L(f)=\beta'^{-1}g\beta$].

The are several comments to make here. Firstly, I would like this to work in the enriched setting as well, so that when $C$ and $D$ are $V$-categories, the three described categories are $V$-categories as well, this means that I shouldn’t mention actual morphisms in the description of $Dual(L,R)$. This oughtn’t be a problem.

Secondly, whilst the third description looks more complicated than the first two it seems to me to sometimes be more natural in examples due to its symmetric nature. To take a very simple example, if you’re describing a Dedekind cut of the rationals, it’s intuitively easier to think of it as a partition of the rationals into two sets $P$ and $Q$ such that every element of $P$ is less than or equal to every element of $Q$, rather than as a single set $P$ with some less obvious condition on it.

Thirdly, following on from that comment, it might be useful to think of the reasonably analogous situation where expressing a vector space $P$ as isomorphic to its own double dual is essentially the same as finding another vector space $Q$ and expressing $Q$ as isomorphic to the dual of $P$ while expressing $P$ as isomorphic to the dual of $Q$.

I should say how the equivalences between the three categories in the theorem are given. The equivalence between $Fix(R L)\subset C$ and $Fix(L R)\subset D$ is just given by the restriction of $L$ and $R$ to these subcategories. The equvialence between $Fix(R L)$ and $Dual(L,R)$ is given on the one hand by $P\mapsto (P,L(P),\eta_P,id_{L(P)})$ and other hand by $(P,Q,\alpha,\beta)\mapsto P$. I will let you guess the other equivalence.

Now to get to the case in hand. We have a category $A$ (which Andrew calls $\mathcal{T}$) – I would really like that to be a $V$-enriched category, but haven’t quite set myself up for that, so we’ll just let it be an ordinary category for now. From that we get an adjunction, Isbell conjugation, between the category of presheaves on $A$ and the opposite category of copresheaves on $A$:

$$L\colon Set^{A^{op}}\to (Set^A)^{op};\quad L(P)(a)\coloneqq Set^{A^{op}}(P,A({-},a))$$ $$R\colon (Set^A)^{op}\to Set^{A^{op}};\quad R(Q)(a)\coloneqq Set^{A}(Q,A(a,{-}))$$

In his comment above, Tom defines $I(A)$ to be either $Fix(L R)$ or $Fix(R L)$, he doesn’t care which.

In his nlab entry, Andrew defines the Isbell duality subcategory of the Isbell envelope of $A$ to be $Dual(L,R)$. I will say a few words about this below.

By the above Theorem, this means that Tom and Andrew were talking about the same thing. And this thing is an extension of the category $A$. By the Yoneda embedding, $A$ naturally embeds fully and faithfully in $Dual(L,R)$:

$$i\colon A\to Dual(L,R);\quad a\mapsto (A(-,a),A(a,-),id,id).$$

Furthermore, an object $W=(P,Q,\alpha,\beta)$ of $Dual(L,R)$ is supposed to be thought of as a “new” object of the category $A$ with its homsets being given by $P$ and $Q$, so for all $b\in A$ we should think “$A(b,W)=P(b)$” and “$A(W,b)=Q(b)$”. Actually this is true in $Dual(L,R)$, namely $Dual(L,R)(i(b),W)\cong P(b)$ and $Dual(L,R)(W,i(b))\cong Q(b)$.

I will explain in another post what this has to do with metric spaces, but here I’ll just say a few words about the way Andrew defines what I’ve called $Dual(L,R)$.

Andrew says that objects in the Isbell duality subcategory of the Isbell envelope of $A$ should consist of a presheaf $P$, a copresheaf $Q$ (or rather $F$ in his notation) and a natural transformation $c\colon P\times Q\to A({-},{-})$ such that two things (specified in Definition 3) are isomorphisms. However, as Todd observes in his helpful MathOverflow answer, the set of natural transformation $Nat\bigl(P\times Q,A({-},{-})\bigr)$ is naturally identified with $Set^{A^{op}}(P,R(Q))$ and $(Set^{A})^{op}(L(P),Q)$. So Andrew’s $c$ gives rise to both my $\alpha$ and $\beta$, his condition of “satisfying Isbell duality” is precisely that $\alpha$ and $\beta$ are isomorphisms. So his category is the same as mine.