Meeting 31
Monday, 19 June 2023
Hosted by the University of Leeds
Supported by the London Mathematical Society
Schedule:
 13:0014:00 Markus Szymik (Sheffield).

14:00  14:30: Coffee break.

14:30—15:30 Chiara Sarti (Cambridge)

15:30  16:00 Coffee break.

16:00  17:00 Adrian Miranda (Manchester.)
Location:
The meeting will be held in the School of Mathematics at the University of Leeds (A link to the campus map is here).
Titles and Abstracts:
Speaker: Markus Szymik
Title: Categorical aspects of racks and quandles
Racks and quandles are algebraic structures related to groups and symmetries. These concepts were rediscovered many times and have found applications across algebra, number theory, geometry, and topology. This talk shall be a friendly introduction to the categories of racks and quandles. The categorical bias brings problems. I will present solutions to some and mention others that are interesting.
Speaker: Chiara Sarti
Title: Posetal Diagrams for LogicallyStructured Semistrict Higher Categories
This talk will present some recent work with my supervisor Jamie Vicary.
We now have a wide range of proof assistants available for compositional reasoning in monoidal or higher categories which are free on some generating signature. Motivated from categorical physics, we generalize the foundational mathematical formalism of the proof assistant homotopy.io, replacing the conventional notion of string diagram as a geometrical entity living inside an ncube with a posetal variant that allows exotic branching structure. We show that these generalized diagrams have richer behaviour with respect to categorical limits, and give an algorithm for computing limits in this setting, with a view towards future applications.
Speaker: Adrian Miranda
Title: Weakinterchange based semistrictification in low dimensional higher category theory
Given a bicategory, its strictification can be described via a presentation in which the only relations are on 2cells, or cells of the highest dimension. The freeness of the underlying category of of the strictification allows pseudofunctors to also be strictified to 2functors; maps between 2categories which preserve all operations of their domain on the nose. On the other hand, pseudonatural transformations cannot be made to respect the globular structure of their domain 2categories on the nose. Via enrichment, this weakness precisely corresponds to the weakness present in a semistrict model of threedimensional categories known as a Graycategory.
We will describe similar semistrictification constructions for tricategories, presenting the example of fundamental trigroupoids of topological spaces. Once again, the underlying data of codimension one in the semistrictification of any tricategory is free on the appropriate kind of generating data, i.e. a 2computad. We use this to extend semistrictification to higher dimensional maps between tricategories, or (3, k)transfors, finding that trihomomorphisms strictify completely but that trinatural transformations only partially strictify. The resulting semistrict trinatural transformations fail to be closed under composition, but upon closing them under composition we are able to form a closed structure on GrayCat, analogous to Gray's closed structure on 2Cat. This is used to describe the homtriequivalences of what is, conjecturally, a semistrictification tetraadjunction. We then consider certain wellbehaved categories enriched over the closed structure on GrayCat, and use them to describe a construction which maps any weak fourdimensional category, or tetracategory, T to a semistrictified structure T'. By construction there will be a tetrahomomorphism from T to T', and we conjecture that it is a tetraequivalence.