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.