Blog - Evan Patterson

How to prove equations using diagrams, part 2: Weak equivalences and e-diagrams

June 10, 2025

In this second part of a series about diagrammatic reasoning, inspired by e-graphs, we review the comprehensive factorization of a functor and propose a definition of “e-diagram” based on initial functors.

How to prove equations using diagrams, part 1: Initial functors

May 27, 2025

I explain how to represent equations using category-theoretic diagrams and how the valid manipulations of such diagrams are described by initial functors. This post is the first in a series about the categorical, operational semantics of e-graphs.

Graded categories as double functors

May 1, 2025

I explain how a category graded by a monoidal category can be viewed as a double functor out of the delooping of the monoidal category. A few consequences and a series of examples are then presented.

Tech, oligopoly, and oligarchy

January 1, 2025

Going into 2025, the tech industry, especially big tech, advances the oligarchic turn in U.S. politics.

Highlights in metal, 2024

December 26, 2024

Continuing the series started last year, I give you my favorite metal albums released in 2024.

Toward compact double categories: Part 2

June 24, 2024

We propose a definition of a compact double category, intended to axiomatize dualities such as the opposite category and the opposite ring. The definition uses the “twisted” Hom double functor introduced in the previous post.

Toward compact double categories: Part 1

June 20, 2024

What is the abstract structure of an opposite category or opposite ring? We introduce some ideas toward axiomatizing such structure as a compact double category.

Algebras are promonads

January 29, 2024

An interesting analogy between algebras over a ring and promonads on a category is formalized using the apparatus of double theories.

Why double categories? Part 1

January 15, 2024

The first of a series of posts on “why double categories?” beginning with the answer that double categories reconstruct the algebra of relations from universal properties.

Highlights in metal, 2023

December 31, 2023

For those with the inclination, my favorite metal albums released in 2023.

Retrotransformations

October 20, 2023

Retrotransformations between lax double functors are introduced as the “multi-object” analogue of a cofunctor between categories. Notions of “monoidal cofunctor” between monoidal categories and of “multicofunctor” between multicategories are then derived as special cases.

Cartesian double theories

October 13, 2023

Cartesian double theories are a new framework for doctrines based on double-categorical functorial semantics.

Unbiased monoidal categories are pseudo-elements

August 15, 2023

Categorifying the observation that monoids are generalized elements of multicategories, we show that unbiased pseudomonoids, such as unbiased monoidal categories, are “pseudo-elements” of 2-multicategories.

Structured cospans as a cocartesian equipment

March 15, 2023

The theory of structured cospans is dramatically simplified by the use of double-categorical universal properties. Specifically, we show that structured cospans form a cocartesian equipment, a result that is stronger yet easier to prove than the usual result that they form a symmetric monoidal double category.

Decorated cospans via the Grothendieck construction

May 30, 2022

Building on the double Grothendieck construction introduced last time, we explain how decorated cospans are instance of the Grothendieck construction. This perspective suggests a natural generalization of decorated cospans, which we illustrate through several examples.

Grothendieck construction for double categories

May 23, 2022

What is the Grothendieck construction for double categories? We explore one possible answer to this question, based on the perspective that double categories are categories internal to Cat. In fact, we suggest a general procedure for doing the Grothendieck construction on any structure that is defined internally to Cat.

Book review: Dynamical Biostatistical Models by Commenges and Jacqmin-Gadda

December 29, 2020

Despite their natural affinity, the statistical and mechanistic traditions of scientific modeling are often poorly integrated. I review a textbook, Dynamical Biostatistical Models, that takes dynamical and mechanistic models seriously.

Classic style and mathematical writing

September 5, 2019

Classic style is an elegant mode of expression that values clarity and avoids self-consciousness. It shares an intriguing connection with the style of mathematical writing.

Book review: Representation and Invariance of Scientific Structures by Patrick Suppes

January 9, 2019

Sixty years ago, Patrick Suppes realized that the notion of a model of a logical theory, so essential to mathematical logic, applies equally well to models in science. I review his final book, Representation and Invariance of Scientific Structures, on the use of formal models in science.

Elements of the scientific stance

November 6, 2018

Scientists, and science itself, have been variously accused of being reductionistic, formalistic, atheistic, and imperialistic. Although charges of scientism are occasionally merited, critiques of science often confuse the metaphysical principles of philosophy with the far milder methodological principles observed by scientists. I explain the difference, distinguishing several flavors of reductionism, naturalism, and other “-isms.”

Book review: Indiscrete Thoughts by Gian-Carlo Rota

August 14, 2018

First published in 1997, Gian-Carlo Rota’s Indiscrete Thoughts is now a minor classic on the culture of mathematics. It is witty and irreverent and difficult not to enjoy. It is also often thoughtful and insightful. Nonetheless, I think its more overtly philosophical parts are seriously flawed. I try to both summarize and critically review this distinctive book.

The role of abstraction in applied math

July 10, 2018

Why is mathematics so difficult to understand and communicate? Mathematicians and nonmathematicians alike often lay the blame on excessive abstraction. I argue instead that abstraction is essential to the mathematical process, even in applied mathematics, and that it need not be a barrier to comprehension by nonspecialists.

The R programming language: The good, the bad, and the ugly

June 15, 2018

The venerable programming language R has gained a new lease on life through the resurgence of data science. Based on my experience as a user, a package developer, and a creator of program analysis tools, I critically evaluate the R language and ecosystem—the good, the bad, and the ugly.

Why is academic writing so bad?

June 1, 2018

According to the stereotype, academic writing is at turns dry, jargony, esoteric, discursive, self-conscious, inward-looking, and—worst of all—just plain incomprehensible. The purpose of writing is to communicate ideas clearly and concisely, but academic writing achieves the opposite. In short, academic writing is bad. Every researcher knows there is some truth to this stereotype but also plenty of exceptions. So why is academic writing often so bad, and what distinguishes the good writing?