https://www.epatters.org/posts/ Writings (mostly) about the mathematical and computational sciences en-us Copyright 2025 Evan Patterson https://validator.w3.org/feed/docs/rss2.html 60 Mon, 04 May 2026 11:30:11 PST https://www.epatters.org/post/tech-and-impunity In the wake of the brutal killings of Renée Good and Alex Pretti, I reflect on how America’s government and tech elites have made common cause in rejecting as illegitimate any constraints on their behavior.

]]> tech-and-impunity Sat, 24 Jan 2026 00:00:00 PST https://www.epatters.org/post/2025-albums In the third installment in this series, I give you my favorite metal albums released in 2025.

]]> 2025-albums Sun, 28 Dec 2025 00:00:00 PST https://www.epatters.org/post/e-graphs-2 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.

]]> e-graphs-2 Tue, 10 Jun 2025 00:00:00 PST https://www.epatters.org/post/e-graphs-1 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.

]]> e-graphs-1 Tue, 27 May 2025 00:00:00 PST https://www.epatters.org/post/graded-categories 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.

]]> graded-categories Thu, 01 May 2025 00:00:00 PST https://www.epatters.org/post/tech-and-oligarchy Going into 2025, the tech industry, especially big tech, advances the oligarchic turn in U.S. politics.

]]> tech-and-oligarchy Wed, 01 Jan 2025 00:00:00 PST https://www.epatters.org/post/2024-albums Continuing the series started last year, I give you my favorite metal albums released in 2024.

]]> 2024-albums Thu, 26 Dec 2024 00:00:00 PST https://www.epatters.org/post/compact-double-categories-2 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.

]]> compact-double-categories-2 Mon, 24 Jun 2024 00:00:00 PST https://www.epatters.org/post/compact-double-categories-1 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.

]]> compact-double-categories-1 Thu, 20 Jun 2024 00:00:00 PST https://www.epatters.org/post/algebras-are-promonads An interesting analogy between algebras over a ring and promonads on a category is formalized using the apparatus of double theories.

]]> algebras-are-promonads Mon, 29 Jan 2024 00:00:00 PST https://www.epatters.org/post/why-double-categories-1 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.

]]> why-double-categories-1 Mon, 15 Jan 2024 00:00:00 PST https://www.epatters.org/post/2023-albums For those with the inclination, my favorite metal albums released in 2023.

]]> 2023-albums Sun, 31 Dec 2023 00:00:00 PST https://www.epatters.org/post/retrotransformations 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.

]]> retrotransformations Fri, 20 Oct 2023 00:00:00 PST https://www.epatters.org/post/cartesian-double-theories Cartesian double theories are a new framework for doctrines based on double-categorical functorial semantics.

]]> cartesian-double-theories Fri, 13 Oct 2023 00:00:00 PST https://www.epatters.org/post/unbiased-pseudomonoids 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.

]]> unbiased-pseudomonoids Tue, 15 Aug 2023 00:00:00 PST https://www.epatters.org/post/structured-cospans-cocartesian-equipment 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.

]]> structured-cospans-cocartesian-equipment Wed, 15 Mar 2023 00:00:00 PST https://www.epatters.org/post/decorated-cospans-via-grothendieck 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.

]]> decorated-cospans-via-grothendieck Mon, 30 May 2022 00:00:00 PST https://www.epatters.org/post/double-grothendieck 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.

]]> double-grothendieck Mon, 23 May 2022 00:00:00 PST https://www.epatters.org/post/review-dynamical-biostatistical-models 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.

]]> review-dynamical-biostatistical-models Tue, 29 Dec 2020 09:00:00 PST https://www.epatters.org/post/classic-style-and-mathematical-writing 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.

]]> classic-style-and-mathematical-writing Thu, 05 Sep 2019 00:00:00 PST https://www.epatters.org/post/review-suppes-representation-invariance 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.

]]> review-suppes-representation-invariance Wed, 09 Jan 2019 09:00:00 PST https://www.epatters.org/post/scientific-stance 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.”

]]> scientific-stance Tue, 06 Nov 2018 09:00:00 PST https://www.epatters.org/post/review-rota-indiscrete-thoughts 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.

]]> review-rota-indiscrete-thoughts Tue, 14 Aug 2018 09:00:00 PST https://www.epatters.org/post/abstraction-in-applied-math 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.

]]> abstraction-in-applied-math Tue, 10 Jul 2018 09:00:00 PST https://www.epatters.org/post/r-lang 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.

]]> r-lang Fri, 15 Jun 2018 09:00:00 PST https://www.epatters.org/post/academic-writing 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?

]]> academic-writing Fri, 01 Jun 2018 09:00:00 PST