The ATCAT Seminar is a seminar on topics in category theory, running since 1973.
The current ATCAT organizer is Cameron Krulewski.
Regular time: Tuesdays at 2:30 PM Atlantic time
Location: Chase Building, Room 319
This is a hybrid seminar. Email ckrulewski [at] dal [dot] ca to be added to the seminar mailing list and to get the Zoom link.
January 13: Robert Paré, Dalhousie University
Categories for the lazy mathematician
Strict double categories are category objects in Cat, and much of their basic properties follow from the general theory of category objects, where the results are proved using standard, albeit complicated, limit arguments. When proving such results, we secretly think that we are working with sets. Now, Cat has many set-like properties, but it is not Set, so we shouldn't be surprised when some things don't work. For example, profunctors don't compose properly.
Instead of pretending that categories are sets, we turn this on its head and pretend that sets are categories. Well, not sets, but the next best thing: the objects of a nice topos. Following Verity, we embed Cat in a topos that closely resembles it, whose objects we call lazy categories.
We investigate how much category theory carries over and what the advantages are of doing this.
January 20: Adrien DeLazzer Meunier, Dalhousie University
An introduction to reconstruction theory
Reconstruction theorems are a key tool in representation theory and theoretical physics. Details vary depending on the setting, but they all follow the same general idea; given some $\mathcal{V}$-enriched category $\mathcal{C}$, and a "fiber functor "$F\colon\mathcal{C}\to\mathcal{V}$ (these are like generalized forgetful functors), there is a canonical functor from $\mathcal{C}$ to the category of modules over the monoid $\text{End}(F)$. The reconstruction theorems give conditions under which this functor is an equivalence, and respects whatever structure $\mathcal{C}$ is assumed to have. Theorems of this sort are responsible for the zoo of definitions of "Hopf-like" algebras, as their categories of modules are used to characterize various flavours of category found throughout representation theory and physics.
In this lecture-style talk, I will give a sketch of reconstruction in the particularly nice context of fusion categories. The goal will be to understand the category-theoretic formalism—as this is consistent across many versions of reconstruction—and to see where and how the setting-specific details creep in. I will also gloss over a newer tool in the representation theory toolkit, known as de-equivariantization, and show how it can be understood through the lens of reconstruction theory.
January 27: Roman Geiko, UCLA
Invertible Stabilizer Codes
A stabilizer code is the common +1 eigenspace of a set of commuting unitaries drawn from a predetermined alphabet. While the most familiar choice of alphabet is the Pauli group, in this talk I will discuss its various generalizations. I will focus on one-dimensional codes defined as eigenspaces of a maximally commuting set of unitaries. From the many-body physics perspective, these codes are interesting because they can be realized as ground states of invertible Hamiltonians. Conjecturally, invertible many-body states (under suitable equivalence relations) are classified by invertible topological field theories. In this talk I will present a classification of invertible stabilizer codes up to finite-depth quantum circuits in terms of certain L-theory. Based on joint work with G. Shuklin.
February 3: Dorette Pronk, Dalhousie University
Factorization systems for double categories
February 10: Harshit Yadav, University of Alberta
February 24: Lorenzo Riva, Harvard CMSA
March 3: Theo Johnson-Freyd, Dalhousie University and Perimeter Institute
March 10: Martina Rovelli, UMass Amherst
March 17: Scott Wesley, Dalhousie University
March 24: Natalia Pacheco-Tallaj, MIT
March 31: Aaron Fairbanks, Dalhousie University
April 7: Naruki Masuda, Northwestern University
September 9: Luuk Stehouwer, Dalhousie University
On a generalization of dagger compact categories
The notion of a dagger compact category combines duals with an involutive dagger functor and provides a categorical setting for operator algebra and quantum theory. The key example of a dagger compact category is that of finite-dimensional Hilbert spaces. On the other hand, the symmetric monoidal category of finite-dimensional super Hilbert spaces is not dagger compact, yet naturally arises in quantum physics when fermions are present. In this talk I will provide a natural generalization of dagger compact categories to arbitrary rigid symmetric monoidal *n-*categories. The goal of this talk is not to overwhelm you with higher categories, but to spell out the definition in the case n=1 in full detail. The result will be a mild generalization of a dagger compact category that covers super Hilbert spaces as a special case.
September 16: Robert Raphael, Concordia University
DL-closures and 2-3 closures applied to the ring $C^1(R)$
In joint work with Barr and Kennison it was shown that commutative semiprime rings have a DL-closure and a 2-3 closure and that the ring of continuously differentiable real-valued functons is not closed in either sense. Our work is devoted to trying to describe the two closures of this ring. The methods are analytic often using basic ideas from calculus. A useful example sent by Alan Dow is presented.
Joint work with W. D. Burgess