The n-Category Café

Thanks! Happy New Year!

To take an example you might find silly, I think some people love operads to describe ways of sticking things together, but don’t talk about them in terms of categories. And I think there are plenty of computer scientists and engineers and linguists who are interested in compositionality but don’t think in terms of categories. For example, if you search under the word “compositionality”, the top hit is (still)

• Compositionality, Stanford Encyclopedia of Philosophy.

which has a lot about logic and Frege and Montague, and nothing about category theory. The second top hit is

• Principle of compositionality, Wikipedia.

which nicely explains the idea:

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. The principle was never explicitly stated by Frege, and it was arguably already assumed by George Boole decades before Frege’s work.

So, I can imagine all sorts of papers that could appear in Compositionality that don’t mention categories. But we’ll see what actually happens.

I don’t think the title of a journal is more than a vague indication of what the journal is about, and people tend to discover what that is over time. I hope Compositionality stays open to a wide range of papers in the general area vaguely indicated on the journal’s webpage, as long as they’re interesting and solid.

Thanks, that’s enlightening.

I don’t think the title of a journal is more than a vague indication of what the journal is about, and people tend to discover what that is over time.

Yes, I agree. But evidently you have the power to influence what kind of person submits to (and is published by) the journal, thereby shaping the journal’s boundaries.

The first thing that popped into my head as a possible non-categorical example of compositionality was actually completely different from logic or linguistics; it was multiscale modelling. The Wikipedia page contains some generalities about what multiscale modelling is, which make the subject sound rather general, but personally I associate the term with hardcore applied people who presumably wouldn’t dream of submitting to Compositionality.

So I guess the question of what non-category-theory the journal publishes is in principle really wide open — though in practice it’s most likely to be determined by the makeup of the editorial board and the all-important first few issues.

One possibility that comes to mind is game theory, where there can be nontrivial synergetic effects that imply an outcome for three players can’t be broken down into pairwise dependencies. (I wrote about this a bit in chapter 4 of arXiv:1509.02958, but I haven’t revisited it much since then.)

There are also various efforts in information theory to develop measures of multivariate correlation and interdependence, generalizing and refining the Shannon mutual information. I’ve taken a crack at this, too, but the best place to start might be the software by James and Crutchfield that implements a bunch of the ideas. The challenge in this area appears to be that it’s much easier to invent an example that points out a problem with somebody else’s function than it is to invent a function that says meaningful things beyond a limited set of toy examples.

Article 1-1 in Compositionality is van de Wetering’s piece on reconstructing quantum theory from new axioms. This is a thing people have been investigating in one way or another since von Neumann, and many of them do it category-theoretically, but not all.

Blake wrote:

One possibility that comes to mind is game theory, where there can be nontrivial synergetic effects that imply an outcome for three players can’t be broken down into pairwise dependencies.

There’s a lot of work on this sort of thing using categories nowadays in ‘open game theory’. For example:

• Neil Ghani, Jules Hedges, Viktor Winschel and Philipp Zahn, Compositional game theory.

Abstract. We introduce open games as a compositional foundation of economic game theory. A compositional approach potentially allows methods of game theory and theoretical computer science to be applied to large-scale economic models for which standard economic tools are not practical. An open game represents a game played relative to an arbitrary environment and to this end we introduce the concept of coutility, which is the utility generated by an open game and returned to its environment. Open games are the morphisms of a symmetric monoidal category and can therefore be composed by categorical composition into sequential move games and by monoidal products into simultaneous move games. Open games can be represented by string diagrams which provide an intuitive but formal visualisation of the information flows. We show that a variety of games can be faithfully represented as open games in the sense of having the same Nash equilibria and off-equilibrium best responses.

Jules helped organize the ACT2019 school, and Neil Ghani spoke at the conference, so there’s definitely an interest in game theory among the people connected to Compositionality.