Details about the Fall 2020 Atlantic Category Theory (ATCAT) Seminar can be found below. Information about previous seminars can be found here.
I will look at the double category of rings with homomorphisms and
bimodules as a motivating example of double category. I will discuss companions
and conjoints. This will lead to “new" morphisms of rings which I think are interesting.
Prerequisites: Some familiarity with rings, modules, and categories.
Most of my talk will be a review of the famous characterization of separable algebras
in terms of dualizability/adjunctibility conditions, and of central simple algebras in
terms of invertibility conditions, in the bicategory of algebras and bimodules. I then
describe my recent generalization of these results in which algebras are replaced by monoidal
(higher) categories.
Prerequisites: some familiarity with the bicategory of algebras and
bimodules, as explained for instance in the talk by Robert Paré.
I will present a generalization of the concept of model category to the context of bicategories as well as a corresponding localization construction. The axioms for a model bicategory are a natural generalization to bicategories of those given by Quillen in the sense that they are obtained by requiring the diagrams to commute up to invertible 2-cells, and by considering a 2-dimensional aspect of the lifting properties which relate these families of arrows (in particular, when we consider a category as a bicategory, the two notions coincide: it will be a model bicategory if and only if it is a model category).
I will define the homotopy bicategory associated to a model bicategory C, whose 2-cells are given by homotopies in C. I will also describe a fibrant-cofibrant replacement for model bicategories, and, time permitting, I will show how we have proved that this yields the localization of C (in the bicategorical sense) at the weak equivalences. Our proof of this result uses a "transport of structure"; the application of this technique in this context is, as far as we know, a novel method.
In [2], my PhD supervisors (Pieter Hofstra and Philip Scott) and I studied the new topos-theoretic phenomenon of isotropy (as introduced in [1]) in the context of single-sorted algebraic theories, and we gave a logical/syntactic characterization of the isotropy group of any such theory, thereby showing that it encodes a notion of inner automorphism or conjugation for the theory. In the present talk, I will summarize the results of my recent PhD thesis, in which I build on this earlier work by studying the isotropy groups of (multi-sorted) quasi-equational theories (also known as essentially algebraic, cartesian, or finite limit theories). In particular, I will show how to give a logical/syntactic characterization of the isotropy group of any such theory, and that it encodes a notion of inner automorphism or conjugation for the theory. I will also describe how I have used this characterization to exactly characterize the ‘inner automorphisms’ for several different examples of quasi-equational theories, most notably the theory of strict monoidal categories and the theory of presheaves valued in a category of models. In particular, the latter example provides a characterization of the (covariant) isotropy group of a category of set-valued presheaves, which had been an open question in the theory of categorical isotropy.
[1] J. Funk, P. Hofstra, B. Steinberg. Isotropy and crossed toposes. Theory and Applications of Categories 26, 660-709, 2012.
[2] P. Hofstra, J. Parker, P.J. Scott. Isotropy of algebraic theories. Electronic Notes in Theoretical Computer Science 341, 201-217, 2018.
We give a finite presentation by generators and relations for the group $O_n(\mathbb{Z}[1/2])$ of $n$-dimensional orthogonal matrices with entries in $\mathbb{Z}[1/2]$. We then obtain a similar presentation for the group of $n$-dimensional orthogonal matrices of the form $(1/\sqrt{2})^k M$, where $k$ is a nonnegative integer and $M$ is an integer matrix. Both groups arise in the study of quantum circuits. In particular, when the dimension is a power of 2 the elements of the latter group are precisely the matrices that can be represented by a quantum circuit over the universal gate set consisting of the Toffoli gate, the Hadamard gate, and the computational ancilla.
The work presented in this talk is motivated by the subadditivity property of maximum degrees of syzygies in a minimal free resolution of a monomial ideal. We give some background for this problem, focus on topological interpretations, and discuss cases where the subadditivity property can be proved from this approach. No prior familiarity with free resolutions will be assumed, they will simply lead us to questions in discrete topology. This talk is based on joint work with Mayada Shahada.
I will first walk through an algorithm of Rabin and Shallit that efficiently solves the four-square Diophantine equation $n=x^2+y^2+z^2+w^2$. The efficiency comes from the use of randomness - there are enough "good seed numbers", so by randomly choosing a number, it is likely that we can hit a good seed that will grow into a solution. Then, I will explain how this algorithm can be adapted to solve the problem of approximating any $2\times 2$ unitary using matrices of a certain kind. The resulting algorithm is a "baby" version of Ross and Selinger's algorithm.