# Atlantic Category Theory Seminar

Information about summer 2022 seminars can be found here.

### Fall 2022 schedule

• September 13rd, 2022
Deni Salja, Dalhousie University

Title: Pseudocolimits of filtered diagrams of internal categories

Abstract:
Pseudocolimits are formal gluing constructions used to combine objects in a category that are indexed by a pseudofunctor. When the objects are categories and the domain of the pseudofunctor is a small filtered category it is known from Exercise 6.6 of Exposé 6 of SGA4 that the pseudocolimit can be computed by taking the Grothendieck construction of the pseudofunctor and then inverting the class of cartesian arrows with respect to the canonical fibration. In my masters thesis I described a set of conditions on an ambient category E for constructing an internal Grothendieck construction and another set of conditions on E along with conditions on an internal category, ℂ, in Cat(E) and a map w : W → ℂ₁ that allow us to translate the axioms for a category of (right) fractions, and construct an internal category of (right) fractions. These can be combined in a suitable context to compute the pseudocolimit of a (filtered) diagram of internal categories.
• September 20th, 2022
Robert Raphael, Concordia University

Title: Applications to two reflectors to the ring C1

Abstract:
This is a progress report on joint work with Walter Burgess of the University of Ottawa.

We procede from work with Barr and Kennison which appeared in TAC 30(2015) 229-304. I will resume this work which concerns embedding a subcategory of rings into a complete one.

The ring C1 of continuously differentiable functions from the reals to the reals is not complete for the two instances that interest us. Thus we seek the completions of C1. This leads us into questions in analysis for which we have progress but lack definitive answers to date.
• October 4th, 2022
Andre Kornell, Dalhousie University

Title: Structured objects in categories of relations

Abstract:
We will look at several examples of biproduct dagger compact closed categories, which are also known as strongly compact closed categories with biproducts. Such a category is a dagger category that is equipped with two symmetric monoidal structures $\oplus$ and $\otimes$, where $\oplus$ is a biproduct and $\otimes$ has dual objects. A biproduct dagger compact closed category is automatically enriched over commutative monoids. In some natural examples, these monoids are cancellative, but in other natural examples they are idempotent. This talk will focus on the latter class.

Specifically, we will look at several examples of biproduct dagger compact closed categories that are enriched over the category of bounded lattices and join-homomorphisms. We will define poset objects and group objects in this setting, recovering a number of familiar examples including discrete quantum groups. Thus, the last part of this talk will focus on the category of quantum sets and their binary relations.
• October 11th, 2022
Robert Pare

Title: The Double Category of Abelian Groups
• October 18th, 2022
Michael Lambert

• October 25th, 2022
TBD

• November 1st, 2022
TBD

• November 15th, 2022
Dongho Lee

• November 22nd, 2022
Sarah Li

• November 29th, 2022
Julien Ross

• December 6th, 2022
Marcello Lanfranchi

Updated August 29, 2022 by Frank Fu