The n-Category Café

I’ll try to give a talk about ‘The foundations of applied mathematics’. Not sure I’ll be ready, but this is one of those things where someone needs to just plunge in and try.

That sounds interesting, John. Apart from anything else, I guess you’ll have to indicate what you consider applied mathematics to be.

A little while before I left Glasgow, there was a surprisingly heated debate within my department about what the boundaries of applied mathematics are. (This came up in the context of teaching.) I took the boundaries to be much wider than several of the applied mathematicians did. For example, as far as I’m concerned, the application of category theory to the modelling of computation obviously deserves to be called applied mathematics, as does the work we were doing in the summer on quantifying biodiversity. I was really surprised to find how strongly some people disagreed.

In the halcyon days before becoming head of department, when I had more time to read, hearing you talk about the search for commonalities behind the kinds of diagrammatic tools here was what most excited me about your project. Will you be pressing on to a general theory of networks?

Grothendieck’s Tohoku paper springs to mind, where he showed that cohomology works perfectly well starting from an abelian category rather than the category of sheaves of abelian groups. In particular, I believe this was motivated by wanting to solve certain concrete problems, and needing the cohomology theories (Weil cohomology, probably) to do it.

Another high point was probably the study by Kan of adjoint functors and Kan extension in the context of simplicial objects. This has huge importance for homotopy theory (and which brings to mind Quillen’s monograph Homotopical Algebra, also massively important).

This paper is helpful:

• Elaine Landry and Jean-Pierre Marquis, Categories in Context: Historical, Foundational, and Philosophical, Philosophia Mathematica 13 (2012?), 1-43.

They emphasize Tôhoku:

Cartan and Eilenberg had limited their work to functors defined on the category of modules. At about the same time, Leray, Cartan, Serre, Godement, and others were developing sheaf theory. From the start, it was clear to Cartan and Eilenberg that there was more than an analogy between the cohomology of sheaves and their work. In 1948 Mac Lane initiated the search for a general and appropriate setting to develop homological algebra, and, in 1950, Buchsbaum’s dissertation set out to continue this development (a summary of this was published as an appendix in Cartan and Eilenberg’s book). However, it was Grothendieck’s Tôhoku paper, published in 1957, that really launched categories into the field. Not only did Grothendieck define abelian categories in that now classic paper, he also introduced a hierarchy of axioms that may or may not be satisfied by abelian categories and yet allow one to determine what can be constructed and/or proved in such contexts. Within this framework, Grothendieck generalized not only Cartan and Eilenberg’s work, something which Buchsbaum had similarly done, but also generalized various special results on spectral sequences, in particular Leray’s spectral sequences on sheaves.

In the context of abelian categories, as defined by Grothendieck, it came to matter not what the system under study is about (what groups or modules are ‘made of’), but only that one can, by moving to a common level of description, e.g., the level of abelian categories and their properties, cash out the claim, via the use of functors, that ‘the Xs relate to each other the way the Ys relate to each other’, where X and Y are now category-theoretic ‘objects’. Providing the axioms of abelian categories 3 thus allowed for talk about the shared structural features of its constitutive systems, qua category-theoretic objects, without having to rely on what ‘gives rise’ to those features. In category-theoretic terminology, it allows one to characterize a type of structure in terms of the (patterns of) functors that exist between objects without our having to specify what such objects or morphisms are ‘made of’. As McLarty points out:

[c]onceptually this [the axiomatization of abelian categories] is not like the axioms for a abelian groups. This is an axiomatic description of the whole category of abelian groups and other similar categories. We pay no attention to what the objects and arrows are, only to what patterns of arrows exist between the objects.

They also mention the rise of adjoint functors and Kan extensions; that will be on my list too.

They somewhat underplay the importance of Quillen’s Homotopical Algebra, perhaps because it only reached its full impact inside category theory rather recently, when homotopical mathematics and $n$-category theory got combined. My talk will certainly cover $n$-categories!

They emphasize the rise of topos theory as an approach not just to algebraic geometry but the ‘foundations’ of mathematics:

In the late fifties and early sixties, it seemed possible to define various mathematical concepts and characterize many mathematical branches directly in the language of category theory and, in some cases, it appeared to provide the most appropriate setting for such analyses. As we have seen, the concepts of functor and the branches of algebraic topology, homological algebra, and algebraic geometry were prime examples. Lawvere took the next step and suggested that even logic and set theory, and whatever else could be defined set-theoretically, should be defined by categorical means. And so, in a more substantial way, he advanced the claim that category theory provided the setting for a conceptual analysis of the logical/foundational aspects of mathematics.

This bold step was initially considered, even by the founders of category theory, to be almost absurd. Here is how Mac Lane expresses his first reaction to Lawvere’s attempts:

[h]e [Lawvere] then moved to Columbia University. There he learned more category theory from Samuel Eilenberg, Albrecht Dold, and Peter Freyd, and then conceived of the idea of giving a direct axiomatic description of the category of all categories. In particular, he proposed to do set theory without using the elements of a set. His attempt to explain this idea to Eilenberg did not succeed; I happened to be spending a semester in New York (at Rockefeller University), so Sammy asked me to listen to Lawvere’s idea. I did listen, and at the end I told him ‘Bill, you can’t do that. Elements are absolutely essential to set theory.’ After that year, Lawvere went to California.

More precisely, Lawvere went to Berkeley in 1961–62 to learn more about logic and the foundations of mathematics from Tarski, his collaborators, and their students. One should note, however, that Lawvere’s goal was to find an alternative, more appropriate, foundation for continuum mechanics; he thought that the standard set-theoretical foundations were inadequate insofar as they introduced irrelevant, and problematic, properties into the picture.

I definitely want to talk about this, but also a bit about the new homotopical/$\infty$-categorical version of topos theory.

It will be extremely easy to talk for at least an hour on these subjects; the hard part will be talking for at most an hour.

John, I’m sure you’ll think of all the obvious “key moments” in category theory, so I tried to think of some non-obvious ones. I came up with two.

The first isn’t really a moment in category theory, but it’s something that perhaps played an important role. It’s Stone duality. In the introduction to Stone Spaces, Peter Johnstone describes it as one of the earliest nontrivial examples of an equivalence of categories. One cannot state the Stone duality theorem in full without mentioning categories. (Well, OK, you could avoid mentioning categories, but it would be long-winded and clumsy.) This provides a good reason for some people to learn basic categorical language. Other dualities provide similar reasons, e.g. Gelfand–Naimark duality, or the duality between affine varieties and finitely generated reduced algebras.

The second one is the realization that category theory could usefully be applied to computer science (e.g. semantics of programming languages). I know very little about the early history of this, but I guess this was the first time that category theory proved itself to be useful outside of pure mathematics.

I think the usefulness of categories in theoretical computer science has also played an important sociological role in the category theory community (not that you’ll necessarily want to mention this). In particular, the fact that many category theorists got jobs in computer science departments provided the community with some stability. As you know, category theory can provoke strong taste-based reactions, both for and against, and at times when it’s been particularly out of fashion within mathematics, the fact that computer science departments have been willing to hire category theorists has surely been important in the continuity of the subject.