Joe Moeller

Algebraic Analysis notes Lecture 1 (7 Jan 2019)

This quarter I’m taking a course called “Algebraic Analysis”. It’s being taught by Carl Mautner, who does rep theory and algebraic geometry here at UCR. I’ve had a bunch of classes with him before, and they were always fun. These are the notes from the first class. It was sort of running through the motivationContinueContinue reading “Algebraic Analysis notes Lecture 1 (7 Jan 2019)”

(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?”

Homeomorphism is not just continuous bijection

A common mistake people make is to think that a continuous bijection is a homeomorphism. This is a reasonable mistake. A bijection is an isomorphism of sets. A bijective homomorphism of groups is an isomorphism of groups. In most algebraic settings I can think of this pattern holds. But it is not true of topologicalContinueContinue reading “Homeomorphism is not just continuous bijection”

Counting the simplex category

I don’t often think about combinatorics, but when I was on a plane recently (coming back from the Netherlands), I started thinking about a question I’ve just kept in the back of my mind for a few years. You can read about the simplex category here. In this blog post I’ll go through the exactContinueContinue reading “Counting the simplex category”

Applied Category Theory in the Netherlands, photo dump

For the past two weeks, I’ve been in the Netherlands for the Applied Category Theory workshop and conference. I stayed in a suburb called Katwijk aan Zee. I ended up working with Pawel Sobocinksi’s group on modelling of open and interconnected systems. I only went to Leiden, so all the pictures are from there. InContinueContinue reading “Applied Category Theory in the Netherlands, photo dump”

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”

Reference List: Graph Products of Groups and Monoids

I made this list because I needed to generalize this concept a bit in my paper Noncommutative Network Models. Graph products of groups were introduced by Elisabeth Green in 1990 in her thesis. The idea was generalized to monoids by António Veloso da Costa in 2001. It’s not a product of graphs, its a “product”ContinueContinue reading “Reference List: Graph Products of Groups and Monoids”