# Foundations Day

In the first talk, Thomas Mikhail will explain how Lawvere's fixpoint theorem gives a uniform treatment of Cantor's diagonal argument, Gödel's incompleteness theorem, Russell's paradox, and the halting problem. In the second talk, Joachim Kock will talk about negative quantities in objective linear algebra (joint work with Jesper Møller), and in particular give meaning to 'the determinant of a finite category'. In the third talk Maxime Ramzi will explain how the nonnegative rationals are the universal recipient for finite cardinalities, not just classically, but surprisingly also in homotopy mathematics.

Everybody is welcome!

The first talk should be accessible to anyone acquainted with the basic notions of category theory; the following talks assume also a bit of homotopy theory.

Programme:

12:00-13:00 Thomas J. Mikhail: *Lawvere's fixed point theorem and its applications*

13:15-14:15 Joachim Kock: *Negative quantities in objective linear algebra*

14:30-15:30 Maxime Ramzi: *Higher homotopy cardinality is groupoid cardinality*

(We finish just in time for the Department Christmas Lunch!)

Abstracts:

Thomas J. Mikhail: *Lawvere's fixed point theorem and its applications*

In 1969 Lawvere published a paper called 'Diagonal Arguments and Cartesian Closed Categories' in which Cartesian closed categories are identified as a suitable general framework to unify diagonal arguments. In this talk we will begin with an overview of Lawvere's fixed point theorem. The remainder of the talk will be dedicated to its applications, including Cantor's original diagonal argument, Russell's paradox, the halting problem as well as Gödel's first incompleteness theorem, the latter being the main focus.

Joachim Kock: *Negative quantities in objective linear algebra*

Objective linear algebra works with slice categories instead of vector spaces and with colimit-preserving functors instead of linear maps. It serves as a method to turn algebro-combinatorial identities into bijective proofs. I will explain an approach to negative quantities in this context, and exemplify its features with an objective treatment of exterior powers and determinants. Groupoids instead of sets are required to encode signs as homotopies. This is joint work with Jesper M. Møller.

Maxime Ramzi: *Higher homotopy cardinality is groupoid cardinality*

Baez-Dolan introduced the notion of groupoid cardinality of a so-called pi-finite space. This is an invariant with values in the semiring of non-negative rationals, and it satisfies a certain number of properties for which it is universal in the discrete world. The goal of this talk will be to explain how, taking cues from algebraic K-theory, one can define the universal such function in a homotopical setting, and outline a proof of the fact that this homotopically universal gadget is nonetheless discrete.