Category Archives: Math

Posets inside of categories

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”

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?”

Algebraic Analysis notes Lecture 11 (4 Feb 2019)

Notes for lecture 10 Last time: for an abelian category A, C(A) is the category of complexes in A. Say $latex f, g \in \mathrm{Hom}_{C(A)} (X, Y)$ are homotopic, f~g, if there are maps $latex s^i : X^i \to Y^i$ such that $latex f^i -g^i = d_Y s + sd_X$. Definition The homotopy category K(A)ContinueContinue reading “Algebraic Analysis notes Lecture 11 (4 Feb 2019)”

Algebraic Analysis notes Lecture 10 (1 Feb 2019)

Notes for lecture 9 Last time: $latex \Gamma : Sh(X; k) \to k-mod$ global sections functor is left exact. We’ll leave sheaves for now to look at derived categories. What do sheaves have to do with cohomology? Poincare Lemma: Let M be a manifold. Consider the following complex of sheaves: where d is the deContinueContinue reading “Algebraic Analysis notes Lecture 10 (1 Feb 2019)”

Algebraic Analysis notes Lecture 8 (28 Jan 2019)

Notes for lecture 7 Last time: we defined additive categories, and kernels for morphisms in additive categories. Definition: The cokernel of a morphism $latex \phi$ (if it exists) is the universal object $latex cok \phi$ with the dual universal property: Definition: an additive category C is abelian if (A4) for any $latex \phi \colon XContinueContinue reading “Algebraic Analysis notes Lecture 8 (28 Jan 2019)”