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Meeting 31

Monday, 19 June 2023

Hosted by the University of Leeds
Supported by the London Mathematical Society

Schedule:

  • 13:00-14:00 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 Logically-Structured 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 n-cube 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: Weak-interchange based semi-strictification in low dimensional higher category theory

Given a bicategory, its strictification can be described via a presentation in which the only relations are on 2-cells, or cells of the highest dimension. The freeness of the underlying category of of the strictification allows pseudofunctors to also be strictified to 2-functors; maps between 2-categories 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 2-categories on the nose. Via enrichment, this weakness precisely corresponds to the weakness present in a semi-strict model of three-dimensional categories known as a Gray-category.

We will describe similar semi-strictification constructions for tricategories, presenting the example of fundamental trigroupoids of topological spaces. Once again, the underlying data of codimension one in the semi-strictification of any tricategory is free on the appropriate kind of generating data, i.e. a 2-computad. We use this to extend semi-strictification to higher dimensional maps between tricategories, or (3, k)-transfors, finding that trihomomorphisms strictify completely but that trinatural transformations only partially strictify. The resulting semi-strict trinatural transformations fail to be closed under composition, but upon closing them under composition we are able to form a closed structure on Gray-Cat, analogous to Gray's closed structure on 2-Cat. This is used to describe the hom-triequivalences of what is, conjecturally, a semi-strictification tetra-adjunction. We then consider certain well-behaved categories enriched over the closed structure on Gray-Cat, and use them to describe a construction which maps any weak four-dimensional category, or tetracategory, T to a semi-strictified structure T'. By construction there will be a tetrahomomorphism from T to T', and we conjecture that it is a tetraequivalence.