Categorical Algebra with Segal Conditions

Abstract

There are many ways to present algebraic structures categorically: monads, Lawvere theories, limit sketches, and more. In this talk, we’ll learn about Yet Another Way to Present Algebra (YAWPA): Segal conditions and Chu and Haugseng’s algebraic patterns.
The basic idea behind algebraic patterns is that in many cases of categorical algebra we have a notion of pasting diagram which tells us how we are going to compose things. These pasting diagrams are made out of elementary features — things like boxes and wires — and can “swallow other diagrams whole” by putting subdiagrams into boxes. These two kinds of morphisms, diagram inclusions and “swallowings”, organize into a factorization system on a category of pasting diagrams. Presheaves on this category of diagrams which send any pasting diagram to the limit over all its elementary subdiagrams is an algebra composing according to those pasting diagrams; that single axiom is the so-called “Segal condition”.
We’ll see how this works for categories and double categories in particular, and, time-permitting, see how to derive the notion of “virtual double category” from the algebraic pattern for categories and muse about the appropriate “virtual triple categories” which are similarly derived from the algebraic pattern for double categories.