## April 5, 2013

### Category-Theoretic Foundations in Irvine

#### Posted by John Baez

I’d like to remind you of this workshop, which is coming up soon:

The schedule of talks is available now; I’ll just tell you what they’re about.

Here are the talks:

Speaker: Samson Abramsky, Oxford

Title: Category theory as a tool for making models

Abstract: I will discuss category theory in its aspect as a tool for  mathematical modelling in the sciences. It has played an important part in computer science and some areas in theoretical physics for some decades, and is being used increasingly in areas ranging from quantum information and foundations to natural  language semantics and game theory. I shall discuss how the  conceptual and foundational features of category theory have a strong influence on its use in mathematical modelling and applications.

Speaker: John Baez, UC Riverside

Title: The Foundations of Applied Mathematics

Abstract: Suppose we take "applied mathematics" in an extremely broad sense that includes math developed for use in electrical engineering, population biology, epidemiology, chemistry, and many other fields.  Suppose we look for mathematical structures that repeatedly appear in these diverse contexts - especially structures that aren't familiar to pure mathematicians.  What do we find?   The answers may give us some clues about the concepts that underlie the most applicable kinds of mathematics.  We should not be surprised to find some category theory hiding here!

Speaker: Olivia Caramello, Cambridge

Title: Grothendieck toposes as unifying 'bridges' in Mathematics

Abstract: I will present a novel view of Grothendieck toposes as unifying spaces in Mathematics being able to effectively serve as 'bridges' for transferring concepts and results across distinct mathematical theories. This approach, first emerged in the context of my Ph.D. research, has already generated many applications into different mathematical fields, including Topology, Algebra, Geometry, Functional Analysis, Model Theory and Proof Theory, and the potential of this theory has just started to be explored. In the lecture, I will explain the fundamental principles that characterize my view of toposes as unifying 'bridges', and illustrate the technical usefulness of these methodologies by discussing a few selected applications.

Title: Category theory and set theory: examples of their interaction

Abstract: I will begin by reviewing basic arguments and counterarguments about either set theory or category theory as a possible foundational theory for mathematics. I will then devote the main part of my talk to a few examples of positive interaction between both theories in the field of logic. Two examples will be considered more specifically, the first one in connection with algebraic set theory, the second one in connection with sketch theory.

Speaker: Hans Halvorson and Dimitris Tsementzis, Princeton

Title: Structuralist foundations for abstract mathematics

Abstract: As Paul Benacerraf famously pointed out, set-theoretic foundations provide too much information for the working mathematician.  Our talk is driven by the question: is it possible or desirable to develop a foundational framework that omits the irrelevant and distracting information?  We consider one proposal motivated by the slogan that, "all mathematically relevant properties are structural."  Unlike many previous structuralist proposals, this proposal is linguistic (syntactic) in nature.  Namely, we present a language L together with an intended semantics, such that the following hold: (1) L is rich enough to be considered foundational; (2) L satisfies the appropriate adequacy conditions with respect to its intended semantics; and crucially (3) well-formed sentences in L are necessarily invariant under isomorphisms of the relevant type of structure.  This proposal is strongly influenced by Michael Makkai’s "structuralist foundation of abstract mathematics," and his language FOLDS (first-order logic with dependent sorts).

Speaker: Ralf Krömer, Siegen

Title: "Psychological priority" vs. effects of training. The foundational debate on Category theory revisited

Abstract: The paper develops a particular answer to a standard objection (by Feferman) against uses of Category theory as a foundations of mathematics. This objection roughly says that concepts like "collection" and "operation" are "psychologically prior" to the concept of category. Our answer amounts to showing the relevance of mathematical training to the foundational debate on Category theory.

First, we briefly review this debate from the beginnings to Bénabou's 1985 proposal to use elementary portions of Category theory as a foundation of "naive" Category theory as a whole; we discuss both some of the relevant mathematical and metamathematical facts and a number of records of how workers in the field interpret these facts and their philosophical impact. The main thesis of the paper is that "psychological priority" (Feferman) is in the last analysis irrelevant for the problem of a "proper presentation" (Isbell) of Category theory; one needs to take into account the effects of training because without training one is not able to judge the "properness''.

In order to develop this claim more fully, a sidestep is made to the epistemology of C.S. Peirce, especially to his account of intuition and the role of consciousness in hierarchies of cognitions. What is particularly relevant for us in Peirce's anticartesian approach is his criticism of the role of the individual for epistemology. While traditional philosophy of the foundations of mathematics seems to seek a foundational stance accessible for every individual in isolation (and this is also implicit in Feferman's conception of psychological priority), Peirce stresses the importance of the presence of a community, of collective processes of regulation of work with concepts, and of learning in such a situation. On these grounds, a (still sketchy) comparison between hierarchies of cognitions in the sense of Peirce and hierarchies of mathematical concepts is presented. The Peircean conception of intuition is found to be relevant whenever the role played by mathematical training for mathematical intuition is stressed. This ultimately leads to a revision of the notion of foundations of mathematics.

Title: Some elements of the descent theory, from the point of view of Algebraic Geometry, with philosophical remarks.

Abstract: In the framework of CT descent theory is a characterization of morphisms (descent morphism) that constitute a theory of generalization and of extension concerning very different mathematical concepts and theoretical points of view.

Also, right before the conference, I’m giving a department colloquium talk on “Key Moments in Category Theory” on Friday May 3. The Department of Logic and Philosophy of Science at Irvine has been doing a number of other category-theoretic activities lately, including Colin McLarty’s visit last spring and Jim Weatherall’s category theory course this spring.

Posted at April 5, 2013 12:06 AM UTC

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### Re: Category-Theoretic Foundations in Irvine

Will there be slides, or even better a video, from the colloquium talk?

Posted by: Emily Riehl on May 1, 2013 1:37 AM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

Sorry, no slides, since I decided a blackboard talk will convey to these philosophers the spirit of how mathematicians do category theory better than slides would: it involves drawing lots of diagrams in real time. And I don’t think they videotape the colloquia, either. But you’re not missing much: I’m sure you know everything I’ll say.

Posted by: John Baez on May 1, 2013 5:04 AM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

From Halvorson and Tsementzis’s abstract

This proposal is strongly influenced by Michael Makkai’s “structuralist foundation of abstract mathematics,” and his language FOLDS (first-order logic with dependent sorts).

What is known about the relationship between Makkai’s foundations and Voevodski’s univalent foundations or homotopy type theory?

The nLab entry FOLDS says

Formally speaking, FOLDS is “merely” a “first-order fragment” of dependent type theory, but its special treatment of equality, its notions of equivalence, and its relationship to higher-categorical structures distinguish it from DTT in general.

But presumably they doesn’t distinguish it from homotopy type theory. So can we simply say

FOLDS is “merely” a “first-order fragment” of univalent foundations/homotopy type theory?

I’m coming increasingly to believe that in the choice of formal system one should look to have as many signs of naturalness as possible. It’s going to be hard to compete with homotopy type theory since it’s such a natural blend of homotopic mathematics and categorical type theory. And looking at applications, does FOLDS allow anything like the treatment of gauge theory we see here?

Posted by: David Corfield on May 1, 2013 10:20 AM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

presumably they doesn’t distinguish it from homotopy type theory

Actually, they do. Both FOLDS and HoTT have a special treatment of equality, a notion of equivalence, and a relationship to higher-categorical structures, but all three of them are different!

FOLDS is not really a foundational system for mathematics. It’s more like a metalanguage in which to talk about general higher-categorical structures.

Posted by: Mike Shulman on May 2, 2013 8:54 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

Did you attend the other talks? If so, anything worth reporting?

Posted by: David Corfield on May 8, 2013 8:45 AM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

I was especially impressed by Olivia Caramello’s talk on topos theory and Brice Halimi’s talk on sketches: I learned some new math from these (and from reading some of Caramello’s papers). I unfortunately missed the very last talk, Hans Halvorson and Dimitris Tsementzis, speaking on a language that “forbids evil thoughts”:

Abstract: As Paul Benacerraf famously pointed out, set-theoretic foundations provide too much information for the working mathematician. Our talk is driven by the question: is it possible or desirable to develop a foundational framework that omits the irrelevant and distracting information? We consider one proposal motivated by the slogan that, “all mathematically relevant properties are structural.” Unlike many previous structuralist proposals, this proposal is linguistic (syntactic) in nature. Namely, we present a language L together with an intended semantics, such that the following hold: (1) L is rich enough to be considered foundational; (2) L satisfies the appropriate adequacy conditions with respect to its intended semantics; and crucially (3) well-formed sentences in L are necessarily invariant under isomorphisms of the relevant type of structure. This proposal is strongly influenced by Michael Makkai’s “structuralist foundation of abstract mathematics,” and his language FOLDS (first-order logic with dependent sorts).

If Mike Shulman attended this talk, perhaps he can say the most interesting thing that wasn’t in the abstract.

Posted by: John Baez on May 8, 2013 3:26 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

I posted this very nontechnical comment on Caramello’s talk on Google+, just to get nonexperts interested in topos theory, and we had a good discussion over there:

At the Category-Theoretic Foundations of Mathematics Workshop at U.C. Irvine this weekend, I met Olivia Caramello, who is a powerful advocate of topos theory as an approach to unifying mathematics. She was a student of one of the best topos theorists in the world, now she’s a research fellow at Cambridge University, and she lives and breathes math. I always enjoy meeting brilliant young mathematicians who are eager to explore mind-blowing new realms and confident in their power to do it. They can be scary, but they’re fun to see — sort of like a lion or tiger.

I’m not going to tell you what a topos is; I’ll just say some stuff about them. They were invented by Grothendieck in the 1960’s as part of his quest to prove some conjectures in number theory. The solutions of a bunch of polynomial equations give, indirectly, a topos, and he thought of this topos as a generalization of a space, hence the name ‘topos’. Later he and other realized that a topos is like a mathematical universe, since the world of set theory is one topos, but there are many more. So, you can think of topos theory as a grand generalization of set theory.

But then people realized that a topos is also like a theory. For many familiar mathematical gadgets - groups, rings, etcetera - there is a topos called the classifying topos of that gadget, which acts like an axiomatic theory of that gadget. In particular, any particular instance of that gadget in the world of sets - for example, any particular group - gives a map from the topos of sets to the classifying topos of that gadget, and conversely.

Olivia Caramello is deeply focused on classifying topoi and how they can be used as ‘bridges’ to relate seemingly different fields of math. She has an easy-to-read introduction to this idea here:

My friend Colin McLarty, who also attended this conference, has written a nice history of topos theory that gives a good flavor of what it’s about, without revealing the — somewhat terrifying, at first — definition of a topos:

It’s taken me a long time to learn what little I know about topos theory — in part because I’m interested in lots of other things, and in part because I found it a bit intimidating for the first few decades. A long time ago, I wrote this introduction:

whose main virtue is that it points you to 2 free books on the subject: the easy one by Goldblatt, and the vastly more profound book by Barr and Wells. But now I’ve reached the point where topos theory makes a lot more sense! So, I should probably expand this page a bit someday.

If you know topos theory and/or a fair amount of logic, you can get a good intro to Caramello’s work here:

However, this is not for the faint-hearted! Here’s a sample, just so you can decide if this is for you:

Now, the fundamental idea is the following: if we are able to express a property of a given geometric theory as a property of its classifying topos then we can attempt to express this property in terms of any of the other theories having the same classifying topos, so to obtain a relation between the original property and a new property of a different theory which is Morita-equivalent to it. The classifying topos thus acts as a sort of ‘bridge’ connecting different mathematical theories that are Morita-equivalent to each other, which can be used to transfer information and results from one theory to another. The purpose of the present paper is to show that this idea of toposes as unifying spaces is technically very feasible; the great amount and variety of results in the Ph.D. thesis [2] give clear evidence for the fruitfulness of this point of view and, as we shall argue in the course of the paper, a huge number of new insights into any field of Mathematics can be obtained as a result of the application of these techniques.

(A ‘geometric theory’ is not one about geometry; it’s a technical term for an axiom system obeying certain properties,which Caramello explains. As I hinted, many different kinds of mathematical gadgets can be described using geometric theories. She also explains the concept of “Morita equivalence”.)

In the subsequent discussion Timothy Gowers asked for some concrete, easily understood examples of the insights Caramello was getting from topos theory. We wound up discussing 1) the relation between Gödel’s completeness theorem and Deligne’s theorem on coherent topoi, and 2) Caramello’s new approach to Galois theory based on Fraïssé limits. It would be great if some experts on topos theory could straighten out any of the confusions we had, and provide some more examples.

Posted by: John Baez on May 8, 2013 3:45 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

Well there’s a ringing endorsement of Caramello’s work by Lafforgue here, which I found out about through answers to this MO question.

Posted by: David Corfield on May 8, 2013 4:14 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

Thanks! Caramello said she’s going to Paris for a half-year (?) or so to talk to Lafforgue and Connes. It’ll be great to have a strong supporter of topos theory talking to these influential mathematicians: the subject may finally become fashionable!

Posted by: John Baez on May 8, 2013 4:58 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

I just read to the end. A stunning endorsement.

Lafforgue alludes to the possibility of extending this technique to $(\infty, 1)-toposes$. Since there are analogous classifying (infinity, 1) toposes, there could be exciting times ahead.

Posted by: David Corfield on May 8, 2013 8:34 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

Hans’ talk turned out to be essentially an advertisement for univalent foundations (homotopy type theory). I got the impression that that may have been a change since the writing of the abstract: that they used to believe FOLDS was the way to go, but are now convinced that UF is better.

Posted by: Mike Shulman on May 8, 2013 4:50 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

It’s a pity there wasn’t a talk about FOLDS… but it would be a tragedy if there hadn’t been a talk about univalent foundations at a 2013 workshop on categorical foundations, so I guess that’s good!

I briefly advertised univalent foundations, homotopy type theory and higher topoi in my talks, but I didn’t really explain them. I just made it clear that all the cool kids are talking about them, and gave some links to learn more.

Posted by: John Baez on May 8, 2013 5:02 PM | Permalink | Reply to this

### Re: Category-Theoretic Foundations in Irvine

Well, all the best philosophers are endorsing HoTT these days :) (May 20 to be precise.)

Posted by: David Corfield on May 8, 2013 7:54 PM | Permalink | Reply to this

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