## December 18, 2012

### Category-Theoretic Foundations in Irvine

#### Posted by John Baez

There will be a two-day workshop:

From the conference webpage:

The aim of this 2-day workshop is to provide a forum in which researchers from philosophy, mathematics, computer science, and allied disciplines can discuss the aims and significance of category-theoretic foundations of mathematics. The interdisciplinary character of this workshop provides a unique opportunity to discuss and deliberate upon what is specific to the success of category-theoretic foundations within the various disciplines.

Speakers include:

• Samson Abramsky (Oxford)
• John Baez (UC Riverside)
• Olivia Caramello (Cambridge)
• Brice Halimi (Paris-Ouest)
• Hans Halvorson (Princeton)
• Ralf Krömer (Siegen)
• Jean-Jacques Szczeciniarz (Paris-Diderot)

This workshop is made possible through generous support provided by the Department of Logic and Philosophy of Science at UC Irvine, School of Social Sciences at UC Irvine, Kurt Gödel Society, University of Notre Dame, and REHSEIS team and SPHERE lab at the University of Paris-Diderot.

The event will be preceded by a departmental colloquium talk on Friday May 3 by John Baez (U.C. Riverside). This event is one of a small cluster of recent category-theory related activities organized by the department of Department of Logic and Philosophy of Science. Others include Colin McLarty’s (Case Western) visit last spring as well as Jim Weatherall’s category-theory course this spring.

Attendance is free, but RSVPs are encouraged prior to April 12. For further details, please contact organizer Michael Ernst.

Posted at December 18, 2012 6:45 PM UTC

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### Re: Category-Theoretic Foundations in Irvine

What will you be talking about, John?

Posted by: Todd Trimble on December 18, 2012 7:37 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

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.

Posted by: John Baez on December 18, 2012 8:08 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

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.

Posted by: Tom Leinster on December 19, 2012 12:37 AM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

Interesting. My personal tendency is to describe your work on biodiversity as applied mathematics, but applications of category theory to computation as theoretical computer science. I imagine it depends a lot on how one was brought up!

Posted by: Tom Ellis on December 20, 2012 9:09 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

Tom wrote:

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.

Perhaps you were defining applied mathematics to be mathematics that is applied to something other than mathematics. That would make sense, but it’s not the definition people use in hiring decisions! For example, mathematical physics does not typically fall under the header of ‘applied mathematics’, nor do applications of math to theoretical computer science.

When it comes to hiring people, applied mathematicians have a strong interest in limiting applied mathematics to mean ‘stuff similar to what we do’.

For university administrators, applied mathematics seems to mean ‘mathematics that can get large grants’. The main application of mathematics they’re concerned with is getting money! That is, a ‘grant application’.

I will have to deal with all these issues somehow in my talk—thanks for reminding me!

Posted by: John Baez on December 20, 2012 10:37 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

When it comes to hiring people, applied mathematicians have a strong interest in limiting applied mathematics to mean `stuff similar to what we do’.

I don’t think this is universal. For instance it hasn’t been my experience.

Posted by: Eugene on December 21, 2012 2:53 AM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

Also, this phenomenon isn’t limited to applied mathematicians.

Posted by: Mark Meckes on December 21, 2012 7:05 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

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?

Posted by: David Corfield on December 19, 2012 8:55 AM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

David wrote:

In the halcyon days before becoming head of department, […]

Ah, so that’s why you’re so quiet these days. You probably mentioned this before, but I occasionally wonder what happened to you. Is this a near-permanent fate, or is there hope for a quick recovery?

Will you be pressing on to a general theory of networks?

Yes, my idea is that people who use math in a variety of applications invent mathematical tools they need in a somewhat informal way… and because ‘applied mathematics’ and ‘engineering’ are somewhat devalued by sophisticated pure mathematicians, the concepts underlying these tools haven’t been investigated in a formal way. (Particle physics and string theory, by comparison, has higher status among pure mathematicians, so their tools are formalized and worked into the fabric of pure mathematical thought.)

And, as you guessed, the tools I’m most interested in are diagrams and networks. To some extent what’s going on is the same old thing: string diagrams used to depict morphisms in symmetric monoidal categories. But the fun starts when we go beyond this, and examine the details of various examples!

I got completely absorbed in stochastic Petri nets and chemical reaction networks, because in this case everything I knew about Feynman diagrams turned out to be relevant, but with probabilities replacing amplitudes. It’s easy to sink into the mire of delicious details. But now I want to soar up a bit and present more of a bird’s-eye view… because there are many other kinds of networks present in applied math, and it’s only when we look at all of them together that we’ll get really big new ideas.

Posted by: John Baez on December 19, 2012 4:45 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

With good behaviour, I should be allowed parole in about 8 months.

By the way, are you thinking of coming over for one of your Parisian sojourns next summer?

Posted by: David Corfield on December 19, 2012 4:51 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

I probably won’t go to Paris: Paul-André invited me, but unfortunately Lisa and I are planning to work in Singapore for the next three summers at least, and I’m trying to avoid rapid zig-zagging across the globe. One big exception is that I’ll be going to Sheffield in 2013 from March 25th to 28th: I’m giving a talk on ‘The Mathematics of Planet Earth’ at the British Mathematical Colloquium.

Posted by: John Baez on December 19, 2012 5:17 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

By the way, my colloquium talk will about ‘Key moments in category theory’, or something like that. The idea is to introduce mathematicians to category theory by giving them a quick historical tour with some of the high points indicated: the big conceptual steps forward, or the big exciting applications.

So if anyone here has opinions about what those key moments are, I’d love to hear them!

Posted by: John Baez on December 18, 2012 10:45 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

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).

Posted by: David Roberts on December 18, 2012 11:43 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

Those were the first two examples that sprang to my mind as well!

Posted by: Todd Trimble on December 19, 2012 12:40 AM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

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.

Posted by: John Baez on December 19, 2012 1:39 AM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

The Landry-Marquis article is from 2005.

Posted by: David Corfield on December 19, 2012 9:10 AM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

This is a parenthetic comment: Kan extensions show up in the study of networks. Namely, if you want to consistantly assign phase spaces to networks, you end up with a Kan extension of the identity functor on your category of phase spaces.

Posted by: Eugene on December 20, 2012 7:23 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

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.

Posted by: Tom Leinster on December 19, 2012 8:12 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

In my view, Jim Lambek’s work on the syntactic calculus should count as the first application of category theory to the real world. Although Lambek didn’t make the connection clear in print before the series of papers on deductive categories in 1968ff he must have been aware of it early on: It arose from his work on multilinear algebra with Findlay in the 1950s which is explicitly mentioned in Lambek’s 1961 paper for the symposium on ‘the mathematics of language’. The Roumanian scholar Ana Burghelea was probably the first person to go in print with this story in 1967.
In the mid 1960s computer science kicked in. The proceedings of the 1965 LaJolla conference has a paper on automata theory from Giv’eon that acknowledges discussions with Peter Freyd. This was the first in a line of research that reworked automata theory from the perspective of categorical algebra (Eilenberg-Wright, Budach-Hoehnke, Goguen, Ehrig, Adamek-Trnkova etc.).
At the same time G.Hotz in Germany in a series of papers on formal languages brought forth the connection between rewriting, braid groups, 2-dimensional topology and monoidal categories: here for the first time a graphical calculus is used in category theory well twenty years before the Joyal-Street work on tensor calculus. David Benson contributed a couple of important papers to this research which somehow undeservedly fell into oblivion.
The 70s then put domains, partially-additive categories and intial algebra semantics on the agenda, topics which in one form or another are still with us today.

Posted by: thomas holder on December 20, 2012 9:49 AM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

This is exciting! I hope I can manage to come. Ironically, being in Princeton with the homotopy type theorists this year means I have a lot farther to come to this conference than I normally would. (-:

Posted by: Mike Shulman on December 19, 2012 7:42 AM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

I wonder where philosophy’s relationship with category theory will go. Won’t posterity scratch its head wondering why over so many decades there was such philosophical indifference to it, aside from the few jumping up and down, pointing at it?

Perhaps we never quite found the killer app. Having it underwrite a major transformation in physics would help.

Posted by: David Corfield on December 19, 2012 9:30 AM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

I am not sure whether this is useful.

On the one hand, set theory is a sophisticated mathematical topic for its own sake, and I am not sure whether category theory is helpful there. Especially, having the property that “everything is a set” without urelements is a strength, you can for example have set induction \forall_x((\forall_{y\in x} \varphi(y))\rightarrow\varphi(x))\rightarrow\forall_x\varphi(x) and recursion, which is useful for methods like semantic forcing. You would rather talk about the “category of models of ZFC” (or whatever way this would then be formalized) in set theory.

For mathematics that use set theory merely as a foundation, I am not sure whether the notion of a “set” as such is really useful, since we explicitly want more than “collections”. Probably type theory is more useful in that case.

Posted by: dasuxullebt on December 20, 2012 11:52 AM | Permalink | Reply to this

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