The ATCAT Seminar is a seminar on topics in category theory organized by the Atlantic Category Theory Group, running since 1973.
Virtual participation is available. Please contact the organizer (Hayato Nasu) to be added to the mailing list and receive the Zoom link.
Information about Winter 2026 seminars can be found here.
September 15
Speaker: Robert Paré (Dalhousie University)
Title: The regular completion of \( {\bf Cat} \) and its relation to lazy categories [Slides]
The category of small categories is a relatively nice category. It's locally
finitely presentable (essentially algebraic) and cartesian closed. But
quotients are notoriously nasty. The problem with
unruly quotients is a familiar one from geometry and topology, which
has led to the introduction of schemes, manifolds, and orbifolds among
others.
The introduction of lazy categories was billed as a solution to the
``quotient problem'' for categories, citing as a motivating example,
the fact that \( {\bf Cat} \) is not even a regular category. So, why
not just make it regular rather than going whole hog to a topos?
This was the question asked, not in those precise words, by
Peter Johnstone.
The regular completion of a category with finite limits, introduced
by Carboni et al., is a now classical construction, and has been
worked over by many people (Hu and Tholen, Lack, ...). In this
talk, I will discuss regular categories and the regular completion.
Then I will recall lazy categories and relate them to the regular
completion of \( {\bf Cat}\). Time permitting, I will also discuss
exact categories, and how they are related to the above.
September 22
Speaker: Robert Morissette (Dalhousie University)
Title: Towards negation for double-categorical logic [Slides]
There are several well-known categorical settings for encoding logic (toposes, hyperdoctrines, and bicategories of relations, to name a few), each with their own advantages and drawbacks when applied in different contexts. Double categories represent an under-developed setting in which to do logic which is able to combine advantages of other settings without compounding drawbacks, and with little "complexity tax". In this talk, I will give an overview of how fragments of first-order logic have been brought "into the realm of double categories" via a connection with fibrations, and share some work in progress on efforts to incorporate negation (of various strengths) into this picture.
September 29
Speaker: Tanner Altenkirk (Dalhousie University)
Title: Algebraic L-Theory via Poincare Infinity Categories
Algebraic L-theory was developed in the late 1960's by C.T.C. Wall and others as a means of classifying higher dimensional manifolds via surgery theory. Importantly, given a Poincare space of dimension greater than four, the theory defines a sequence of L-groups that encode the obstructions to surgery, which in turn tells us if the space is homotopic to a manifold. Ranicki formulated these L-groups using chain complexes and showed that the L-groups form a spectrum with four-periodicity.
More recent joint work by Baptiste Calmès, Markus Land, Thomas Nikolaus, and six others reformulates algebraic L-theory using the notion of a Poincare infinity category. The authors show that this reformulation recovers Ranicki's theory, and furthermore it shows that one can do L-theory on any Poincare infinity category, providing a substantial generalization of the theory.
In this talk, I will introduce Ranicki's chain complex formulation of L-theory and outline some challenges associated with it. I will then introduce the infinity categorical formulation of Calmès et al and demonstrate that this approach is not only more general than the original theory, but also more natural.
October 6th
Speaker: Hayato Nasu (Dalhousie University)
Title: Enriched categories in a virtual double category [Slides]
Enriched categories are ubiquitous in mathematics. In this talk, I will introduce the basic idea of what enrichment in a (virtual) monoidal category means, and how a (virtual) bicategory and a (virtual) double category offer us natural generalizations of it. I will show many examples of enrichment in these three settings, and one construction of a new enrichment from an old one, namely the lax slice. In addition, I will explain how we used this construction to recover and enhance the result of Cockett and Garner on restriction categories as enriched categories.
This talk is based on joint work in progress with Nathanael Arkor, Yorgo Chamoun, Bryce Clarke, Vít Jelínek, and Theofilos Tsantilas, which started as the 2026 Adjoint School research project "Double Category Theoretic Perspective on Partiality".
October 13th
No seminar
October 20th
Speaker: Deni Salja (Dalhousie University)
Title: A Complicial Calculus of Fractions
October 27th
Speaker: Jana Nickel (the University of Hamburg)
Title: Relative monads, Kleisli categories, and distributivity
November 3rd
Speaker: Daniel Almeida (the University of Ottawa)
Title: Some categorical aspects of algebra with dependent sorts
November 10th
No seminar (reading week)
November 17th
Speaker: Dorette Pronk (Dalhousie University)
Title:TBA
November 24th
Speaker: Aaron Fairbanks (Dalhousie University)
Title:TBA
December 1st
Speaker: Daniel Teixeira (Dalhousie University)
Title:TBA
No talk on December 8th (Monday schedule) or later this year.