To help answer this question, I've written a little booklet—a collection of expository notes inspired by the 2018 Applied Category Theory workshop that took place in the Netherlands earlier this year. That booklet is now available on the arXiv, and you can access it by clicking the title page below!
The content—which assumes knowledge of basic category theory—is an extended version of a talk I gave at the category theory seminar at Johns Hopkins University earlier this year. Initially, I planned to turn those talk notes into a blog post, but one blog post turned into two, and two turned into five, and five turned into 50 pages, and so here we are!
Now there is much more to applied category theory than what I've included in the booklet. I've only presented a very tiny subset of hand-selected ideas. For that reason, the notes are sprinkled with lots of links to other resources where you can discover more themes, constructions, and examples within the field.
"Category theory is a branch of mathematics originally developed to transport ideas from one branch of mathematics to another, e.g. from topology to algebra. Applied category theory refers to efforts to transport the ideas of category theory from mathematics to other disciplines in science, engineering, and industry."
While in the Netherlands for the ACT workshop, I had some time to explore Amsterdam (and eat lots of stroopwafels). Here's a view of the city, taken from the inside the Amsterdam Public Library. Pardon the window glare!
During the workshop, I worked with Martha Lewis, Jade Master, and Brad Theilman on a project involving translations between language models of the kind formalized by Coecke et. al. (Our results are now posted on the arXiv.) This is a little nook inside of the Lorentz Center where we spent much of our time.