SEMINAR: APPLIED CATEGORY THEORY

MAT 563
Fall Semester 2018
Coordinator: Jonathan Lorand
Meeting place and time: Y27 H12 / Mondays 15:00 - 17:00
Prerequisites: Linear Algebra and Algebra
See the info-sheet for the seminar.


Organizational

IMPORTANT! If you would like to participate in the seminar, please do two things: 1) write me an email, and 2) book the module (MAT 563) on the online platform of the University of Zurich. The first physical meeting will be on Monday, October 1^st; beforehand we will interact online via the forum of the seminar (see below). Participation in the seminar is limited to a maximum of 12 people; I will distribute the first topics via email in advance, so that we are ready to go for the first talk on October 1^st.


Description

The aim of the seminar is to give an introduction to the new field of research called "applied category theory". We will learn the basics of category theory while exploring many connections with areas of application outside of mathematics (e.g. engineering, chemistry, systems biology, computer science, etc.). The main reference will be the new book "Seven sketches in compositionality" by Brendan Fong and David Spivak, which is available here: http://tinyurl.com/7sketches
(Note: the book is under editing until October 15; if you spot any typos, or have other comments/suggestions, please write here!)

The structure of our seminar will be designed to support an interactive and collaborative learning environment. We will all do the same weekly reading, so that we have one common set of material to discuss. Students will be asked to give a seminar presentation, as well as write short weekly reading responses in an online discussion forum. We can learn a lot through interaction with each other!

The main inspiration for designing this seminar comes from my recent participation in this online seminar and subsequent workshop in applied category theory. Recently, one of the participants Tai-Danae Bradley wrote up nice expository notes on applied category theory - have a look! Also, until the end of Summer 2018 John Baez is running an open online course in applied category theory based on the Fong-Spivak book as above. The course is being run through the forum of the Azimuth blog. It is very worthwhile to sign up to that forum and have a look at his lectures and the many discussions happening there.


Further references

A "theory" reference that I would recommend for beginning with category theory is this (freely available) book by Tom Leinster: Basic Category Theory. Among the various further texts, there is, for example, Emily Riehl's (also free) Category Theory in Context, and the classic book Categories for the Working Mathematician by Saunders MacLane.

Another book by David Spivak which is also focused on examples and applications in the sciences is based on this text: https://arxiv.org/abs/1302.6946


Format of the seminar

Each week, we will all read a portion of the book "Seven sketches in compositionality" as part of the "homework". The other part of the weekly homework is to post a short "reading response" in the online forum of the seminar, which will be hosted on GitHub (see below for more on how this will work). Finally, at each meeting of the seminar, someone will do a presentation on the reading that we did the preceding week. The idea is that everyone will have already read the material, and the speaker's job is to summarize and discuss the material that we read, give further insights, explain details, answer questions, and stimulate further discussion.


The online forum

The forum will be hosted on GitHub, which is a website which will allow us to access and edit a LaTeX file (our "forum") simultaneously.
I will send a link to the forum to all seminar participants around the start of the semester. I expect that most of us will not yet be familiar with GitHub. Part of your homework for the first week of the semester is to get set up with GitHub and figure out the minimum of how this works! Don't worry, the internet is full of help (also, I will help as best I can), and I believe the effort required at the beginning is very worthwhile - GitHub is a useful tool. And you'll need it for this seminar!
The other thing you'll need is to know a very basic amount of LaTeX. Also here, don't worry: the internet is full of help, and you will not need to know much. But if you are new to LaTeX, you will need to make a certain effort. Again, worthhile: LaTeX is the common digital language for writing mathematics.

For getting set up with GitHub and LaTeX, I've posted some instructions and links in a section below the schedule.


Preliminary schedule of talks

All page numbers and sections refer to the book "Seven sketches in compositionality", unless otherwise noted.

Date	Speaker	Topic
17.09	*online*	GitHub and LaTeX (no physical meeting!)
24.09	*online*	Introductions on the online forum (no physical meeting!)
01.10	Marija	Sections 1.1 – 1.4: Generative effects, posets
08.10	Edona	Section 1.5: Galois connections
15.10	Marino	Sections 2.1 – 2.3: Symmetric monoidal posets, enrichment
22.10	Nicola	Sections 2.4 – 2.5: Constructions with enriched categories
29.10	Alain	Sections 3.1 – 3.3: Databases, categories, functors...
05.11	Niklas	Sections 3.4 – 3.5: Adjunction, data migration, (co)limits
12.11	Carina	Sections 4.1 – 4.3: Enriched profunctors
19.11	Marija	Sections 4.4 – 4.5: Categorification, compact closed categories
26.11	Olga	Sections 5.1 – 5.3: Props, signal flow graphs
03.12	Maryia	Section 5.4 + exta material: Graphical linear algebra (notes)
10.12	Edona + Marija	Sections 6.1 – 6.3: Colimits, hypergraph categories
17.12	Jonathan	Summary

Getting set up with GitHub.

Some important intial information (read fully before doing anything):

Every one of us will need a GitHub account. You do NOT need to pay for anything - anyone can sign up for a *free* account. To sign up, go to https://github.com/ and find your way there. (Note: see also https://education.github.com/pack ).

GitHub uses so-called "repositories'': this word bascially just means a "container" or "folder" where a bunch of files can be stored and edited. There are respositories that are private, and those that are public. The repository for our forum will be private, and hosted by me (you will not have to set up any repository). Once you are signed up with GitHub, i.e. once you have an account at github.com, I can invite you via GitHub to be a participant in the repository for our forum.

Let me explain briefly how we will use GitHub to run our forum. Online, in our repostory at github.com, there will be a LaTeX file (together with a rendered/compiled PDF version of it, pluse some auxiliary files which may be ignored) which we will all be able to access and edit. For example, each week, after we have read a portion of the book, each of you will post your short reading response in the forum file. Further comments and discussion are also welcome. In order to edit the forum file you will install a software application called "GitHub Desktop" on your own personal computer which will allow your computer to hold a copy of the forum file (and all the files in our respository) and to synchronize this local copy with the "main" file which is on github.com. The software GitHub Desktop can be downloaded for free here: https://desktop.github.com/. (Note: there are other ways to set things up, which do not involve GitHub Desktop, but I would recommend this way as the easiest).

Once you have a GitHub account, once you have been added by me to be a collaborator on the forum's repository, and once you have installed GitHub Desktop, then you can open GitHub Desktop and "clone" the forum repository which is on github.com. This will create a copy of the repository on your computer (you can choose where this folder is stored). I then recommend using the tutorial inside of GitHub Desktop which will teach you quickly the basics of how to use it. Further information and guides can also be found on the webpages of https://desktop.github.com/. Basically, the workflow will be as follows: to make an edit on the forum, you open GitHub Desktop and click the button for "syncing" your local clone of the repository with the one online. Then you open the LaTeX file for the forum using the LaTeX editor on your computer. Once you are finished, compile and save your LaTeX file, and then switch back to GitHub Desktop and "sync" you local version of the file with the "main" file online. Now your edits will apear in the main file and be visible to everyone else! In the rare event that two people happen to edit the same portion of the forum document at the same time, GitHub has a way to help resolve such a so-called "merge conflict". I'll write instructions on how to deal with this directly into the forum file, so you can read it there.

If at any point questions come up that you can't figure out, please ask me and I will try to help.

P.S. At the moment, I don't think GitHub Desktop is available/installed on the computers of the math instute. There seem to be other softwares installed for using GitHub, but I do not know how they work (of course, if you know how, you can use the terminal or the softwares installed). I will try to find out more about this?


Getting set up with LaTeX

If you are unfamiliar with the basic idea of how LaTeX works, please read about it first a bit in the internet, e.g. see https://en.wikipedia.org/wiki/LaTeX

To use LaTeX on your personal computer, there are various software packages (also called "distributions") to choose from. If you do not have such a software package installed already, I recommend navigating to https://en.wikibooks.org/wiki/LaTeX/Installation. Read the short introduction there and then, in the first subsection, titled "Distributions", follow the links for the LaTeX distribution package that is reommended for the operating system that you use. I recommend using default pacakges, and not any custom packages.

For this seminar, you will not need to learn how to set up and format LaTeX documents, since I will already have set up the forum file. You will need to get a rough sense of how LaTeX works in general, and be able to write the mathematics which you wish to use in making your comments in the forum. This can remain fairly simple if you wish, or it can become more complicated as you learn more LaTeX (e.g. how to draw commutative diagrams, embed an image, etc.).

Further information and resources can be found, for example, at https://www.latex-project.org/get/, and on many other helpful sites throughout the web. Tutorials, help-forums, youtube videos, etc. abound!

Note: there are several software packages for LaTeX already installed on the math instute computers (Texmaker, TeXworsk, TeXstudio). However, it is at the moment not clear to me how straightforward it is to set up syncing with GitHub on the institue computers, since I think that GitHub Desktop is not installed.