So I’m participating in this year’s Applied Category Theory conference (with a virtual poster—check it out here!), and the preconference Tutorial Day has just kicked off. This post summarizes my notes from the first lecture given by David Spivak. I’ll keep posting my notes from the conference in the upcoming week or so.

*Quick links:*

Tutorial Day notes:

- Lecture 1: Introduction to applied category theory (David Spivak, video)
- Lecture 2: An introduction to string diagrams (Fabrizio Genovese, video)
- Lecture 3: The Yoneda lemma in the category of matrices (Emily Riehl, video)
- Lecture 4: Monads and comonads (Paolo Perrone, video)

The first lecture is entitled “Introduction to Applied Category Theory,” where David introduced to the audience some basic categorical ideas in nontechnical terms. He began the lecture by roughly describing a category as a network of relations and category theory as a field that thinks math as a whole and attempts to find general patterns that persist throughout math, where math, in his words, encompasses stable thought patterns that can last for a very long time.

Category theorists invent categories all the time. (DS)

Next, David went on to illustrate what category theory is by presenting a very small toy category (a lattice of natural number division), where he introduced a product of $a$ and $b$ as the last thing that goes into both $a$ and $b$, which in the toy example was just the greatest common divisor of two natural numbers. More generally, a product in a preordered set in a greatest lower bound, as Jade Master pointed out in the conference Zoom chat.

a divisibility lattice (not David’s original example but one taken from Wikipedia; credit: Ed g2s)

After the toy example, we were presented with two general remarks about category theory:

- Category theory looks for common patterns throughout math, and beyond [which is where applied category theory becomes relevant].
- Category theory was invented (1940s) to consider bridges between [many fields of math]. In a sense the very project of math itself is being considered in category theory.

To illustrate, David named a few renowned scholars and the fields/concepts they respectively invented:

- Euclid: shapes, lines, geometry
- Newton/Leibniz: rates of change
- Pascal/Fermat: probability
- Boole: formal logic
- Noether: abstract algebra
- Turing: computation

All of math was invented by someone to help them think about some aspect of their world. (DS)

Next, David gave a few more established examples of what category theory can do, including a category of resources and processes (which is discussed in detail in his coauthored book [with Brendan Fong] *Seven Sketches in Compositionality*), a category of algebraic systems of equations (e.g., the two systems of equations [$y - x^2 = 3, y + x = 9$] and [$x = 2, y = 7$] qua categorical objects are connected), and a category of formal logic among others (e.g., the deductions “if A or B then C” and “if A then C and if B then C” are equivalent).

Finally, David perceived category theory from a number of “sociological” angles. To name a few:

- conservative: benefits from wisdom of math already invented
- libertarian: you are free to create as you see fit, working within a background framework that is minimally invasive but highly functional
- progressive: everyone can contribute to making a richer, more meaningful field
- postmodern: no perspective is “right,” and navigating between perspectives lets you look at a problem from all sides
- hippy: it’s all about relationship and connection
- practitioner: used to organize and learn from “big data,” manage complexity of large and evolving software projects…

So it seems category theory does have a pleasing facet to everyone!

*References mentioned in the lecture:*

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