The n-Category Café

“Compositionality” means something different than “compose”. It means “the ability to determine properties of the whole from properties of the parts together with the way in which the parts are put together.” And it’s a common buzzword in engineering and computer science, these days.

From Wikipedia’s Principle of Compositionality:

In mathematics, semantics, and philosophy of language, the principle of compositionality is the principle that the meaning of a complex expression is determined by the meanings of its constituent expressions and the rules used to combine them. This principle is also called Frege’s principle, because Gottlob Frege is widely credited for the first modern formulation of it. However, the idea appears already among Indian philosophers of grammar such as Yāska, and also in Plato’s work such as in Theaetetus${}^{[citation \; needed]}$. Besides, the principle was never explicitly stated by Frege${}^{[1]}$ and it was arguably already assumed by George Boole${}^{[2]}$ decades before Frege’s work.

This concerns compositionality of meaning of an expression, but in engineering or computer science, compositionality now often refers to the behavior of a system. One wants to be able to easily analyze the behavior of the whole in terms of the behaviors of its parts.

So, the word should bring to mind algebras of operads, or functors between categories (as in “functorial semantics”).

“Compositionality” was not my first choice of title, but it’s what most the editors wanted. I wanted “Applied Category Theory”, but that has its own problems.

Luckily, I’m more interested in the journal than its title.

I think the journal wants to stay rather open-minded about the different formulations of compositionality in different fields.

In category theory, compositionality is captured by:

1. Functors, since functors preserve composition: $F(f \circ g) = F(f) \circ F(g)$

2. Monoidal functors, since monoidal functors also preserve tensor products: $F(f \otimes g) \cong F(f) \otimes F(g)$

3. Operad algebras, since these preserve operadic composition: $F(f \circ (g_1, \dots, g_k)) = F(f) \circ (F(g_1), \dots , F(g_k))$

and so on. All these are ways of saying the behavior of a big thing can be computed in a systematic way from the behavior of its parts, given how the parts are stuck together.

So, I would definitely say that some principles of compositionality fits well with the general principles of category theory. But I see no reason why all versions of compositionality have to be captured by simple schemas like 1-3.

What about the recursive property

$$H(\mathbf{p} \circ (\mathbf{q}_1, \ldots, \mathbf{q}_n)) = H(\mathbf{p}) + \sum_i p_i H(\mathbf{q}_i)$$

satisfied by entropy? People who use the word “compositionality” would include that as an instance, I assume…?

And I’m not convinced that there are significant advantages to be gained from onymous refereeing. However, I’m open to being convinced! What do others think?

I’ve often thought it would be very beneficial for the reviewer and author to engage in discussion. Even public discussion as far as possible (at least, say, making parts of the discussion public if the paper is accepted), so that anyone can benefit from the insights gained.

This is no doubt far too idealistic to be able to applied in practise, but I think in principle it would be a good process for helping the dissemination of mathematical ideas.

I agree! In fact, in my experience, the refereeing process often approximates a discussion: the referee points out a problem or makes a suggestion, the author fixes it and/or makes a reply, and iterate a few times. Current journal design means that this way of having a discussion places a high burden on the editor as middleman, but with modern technology there’s no reason that has to be the case. Nor do I see any reason that a discussion between author and referee would have to identify the referee by name.

Mike wrote:

But one thing I’d be interested in hearing more folks’ opinions on is the idea of authors knowing the identity of referees, as discussed on reddit.

People like to argue about this—and not just in this particular reddit thread; in general. Some people think it will be great; others think it will be terrible; others think it will be something in between. In general people’s opinions on this issue seem highly correlated to how much they like, or hate, the existing system.

The arguments tends to be highly theoretical, since don’t have enough data on what actually happens in such an alternative system. Obviously the details matter, but it’s not obvious how they matter.

Given this, it seems worthwhile to let people experiment with alternative refereeing systems on a dual-consent basis… just to find out what actually happens.

I guess the upshot of this remark is that if we do such an experiment, we should ask participants to report their experiences, so we learn something.

Well, I’m sorry for bringing it up again, then. I haven’t been a party to such an argument before, so I’m not familiar with the points that tend to be made. Have any such arguments been had in a public forum that you can post a link to? I’m particularly interested in hearing what advantages people think there will be to onymous refereeing, since I haven’t been able to think of many.

Mike wrote:

Well, I’m sorry for bringing it up again, then.

No problem. It’s certainly more interesting than arguing about the title of the journal, especially since that’s already decided and I can’t do anything about it!

Have any such arguments been had in a public forum that you can post a link to?

I’m having a bit of trouble finding everything I’ve seen, but here are two:

• Bertrand Meyer, End anonymous refereeing, Communications of the Association for Computing Machinery.

• Pros and cons of open peer review, Nature Neuroscience.

The latter is an editorial in favor of anonymous refereeing, written by an anonymous author. It mentions that the British Medical Journal has abolished referee anonymity.

Thanks for the links; I’ll have a look.

we should ask participants to report their experiences, so we learn something.

Yes. Although there are so many indirect effects of a choice like this that it’ll be hard to learn everything that’s relevant, and merely surveying authors and referees as they go through the process could create a false sense of having found all the answers. How does it affect participants’ career trajectories 20 years in the future? How does it affect the ability of editors to find good referees, and the pool of authors submitting to the journal? How does it affect the lives of people who may read a paper and its open reviews but who never submit to or referee for the journal themselves? Etc.

I had a look at your two links. The first one seems to be mainly about computer science, with its universally-acknowledged-to-be-messed-up system of “conference publication”. In his footnote 2 link the author explicitly mentions that

The situation may be different in other fields… In mathematics they can say “your proof is wrong”.

As you mentioned, the second one seems to be mainly addressing mandatory openness, although some of the problems it brings up are relevant to the voluntary case too. It does also mention the results of one previous experiment in voluntary openness:

Peter Strick, editor-in-chief of the Journal of Neurophysiology, notes that when the journal tried encouraging voluntary open review a decade ago, the editors quickly realized that this system promoted more problems than it solved, including bland and cautious reviews. The journal also experienced an occasional breakdown of the peer-review process, in which authors and referees bypassed the editors completely in negotiating how a paper should be revised.

If you or anyone can find a public argument in favor of voluntary open reviewing in mathematics, I’d be especially interested.

“applied category theory” would be the application of categorical ideas … to other fields of mathematics.

In particular, this is how I’ve always understood the journal titles “Theory and Applications of Categories” and “Applied Categorical Structures”. For instance, the policy page of TAC says

The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. [emphasis added]

while the subtitle of Applied categorical structures is

A Journal Devoted to Applications of Categorical Methods in Algebra, Analysis, Order, Topology and Computer Science

Category theorists love terminology, and discussions about terminology. I don’t know snappy terms to make the distinction you’re pointing out, but people sometimes distinguish “applications within mathematics” and “applications outside mathematics”.

We could talk about “internally” and “externally” applied category theory… though I don’t think this is snappy enough to catch on. Then we could have “external applications of internal categories”, etc.