A poset can be thought of as a category with the property that each hom-set is either empty or singleton (technically that’s a preorder, but I’ll skip that for now). There’s also a category of posets. In my work lately, I’ve actually wanted to point at an object in a category and say “this oneContinueContinue reading “Posets inside of categories”
Category Archives: Category Theory
Adjoint School 2023
Category theory has been “applied” in one sense or another since the beginning. Eilenberg himself studied automata theory. But the applied category theory community as it currently exists formally coalesced in 2018. All at once, we had the first instance of the Applied Category Theory conference, the accompanying Adjoint School, and the announcement of theContinueContinue reading “Adjoint School 2023”
Delooping, and internalization vs enrichment
Originally I was planning to write a post called something like “monoid facts everyone should know”, but I’m going easy on myself and giving you just one fact for now. If you ask someone for the definition of a monoid, there are two sorts of answers you’ll get: it’s a set equipped with an associativeContinueContinue reading “Delooping, and internalization vs enrichment”
A different string presentation of monads
My intention with this blog post is not to teach what a monad is, and it’s not to teach how string diagrams work. I just want to share some strings I drew to represent monoidal monads. This post is the first part of a series of posts where I present a diagrammatic language I usedContinueContinue reading “A different string presentation of monads”
Combinatorics, Lecture 1 (26 Sep 2019)
John Baez is teaching a course on combinatorics this quarter. I’m taking detailed notes and texing them up. I’m also going to start blogging them. Credit to Tim Hosgood for the pictures. Prehistory of the course Larry Harper taught this course in the past. John is going to be talking about combinatorial species. He previouslyContinueContinue reading “Combinatorics, Lecture 1 (26 Sep 2019)”
What is the Grothendieck construction like?
This is my best attempt at an intuitive introduction to the Grothendieck construction. I’ll give you the definition, but not before warming up to the idea. I’ll start with the earliest conceptual ancestor I could come up with: addition. Numbers, Addition What am I going to tell you about addition that you don’t already know?ContinueContinue reading “What is the Grothendieck construction like?”
(Co)products and Subcategories
Sometimes the ideas of (co)limits and subcategories don’t really play nicely with each other. A (co)limit of a diagram in a category has two basic pieces. One is an object of the category which people usually think of as being the “result” of taking the (co)limit. The other is a (co)cone, which is a bunchContinueContinue reading “(Co)products and Subcategories”
Pointed category: why is it defined that way?
Before I start talking about what I want, I should point out that pointed category is pretty much the lowest generalization of abelian category, which is an important concept when thinking about algebra from a categorical perspective. What is a pointed category? What should the term ‘pointed category’ refer to? Let’s start with a simpler case.ContinueContinue reading “Pointed category: why is it defined that way?”
Reference List: Categorical Network Theory
By “categorical network theory”, I mean the study of networks or graphs using category theory. A lot of the time, the graphs in these works are morphisms in a category, or 1-cells in some sort of 2-dimensional category. Here are some papers on categorical network theory: 2013 Spivak, The operad of wiring diagrams: formalizing aContinueContinue reading “Reference List: Categorical Network Theory”
Joe Moeller 