### Category-Theoretic Foundations in Irvine

#### Posted by John Baez

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

- Category-Theoretic Foundations of Mathematics Workshop, Department of Logic and Philosophy of Science, U.C. Irvine, May 4-5, 2013.

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.

**Speaker: Brice Halimi, Paris Ouest University (IREPH) & SPHERE**

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.

**Speaker: Jean-Jacques Szczeciniarz, Université Paris Diderot Paris 7**

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.

## Re: Category-Theoretic Foundations in Irvine

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