People / Evan Patterson

Principal Research Scientist

About

Evan Patterson is an applied mathematician, data scientist, and software engineer. He holds a PhD in Statistics from Stanford University and a BSc in Mathematics and Physics from Caltech. Evan aims to build next-generation tools for scientific and statistical modeling, structured by the concepts of category theory. In keeping with the spirit of the Topos Institute, he believes that science and technology form an integrated whole and should be employed to benefit, rather than to exploit, humanity.

Personal website

https://www.epatters.org/

GitHub

@epatters

LinkedIn

Evan Patterson

Blog posts

Changing the frame: Formal modeling with LLM assistance, Part 1

Tools for formal modeling have so far favored generality and lack of constraints over well-adaptedness and safeguards. The emergence of LLMs is changing the balance by making it easier to create formal artifacts but harder to be confident in their reliability. Next-generation technology for…

How to prove equations using diagrams, Part 2

This is the second part in a series about diagrammatic reasoning, inspired by e-graphs. Last time, we reviewed the concept of initial functor and showed by example how to calculate with diagrams and initial functors. This time, we make that calculus more systematic and we reconceive e-graphs in…

Growing the Topos tech team

Mathematics and science achieve their greatest impact when innovative basic research is coupled with translation into robust and usable technology. That is easier said than done. Today, research and development, despite appearing conjoined in the acronym “R&D,” are rarely done together. Research…

How to prove equations using diagrams, Part 1

An e-graph, short for “equality graph,” is a data structure that maintains a congruence relation on expression trees: an equivalence relation stable under forming new expressions. First devised by Nelson and Oppen in 1980 (Nelson 1980; Nelson and Oppen 1980), e-graphs received a surge of new…

Graded categories as double functors

At last week’s Topos Colloquium, Rory Lucyshyn-Wright told us about categories graded by a monoidal category, following his recent preprint (Lucyshyn-Wright 2025). Graded categories, short for locally graded categories, were first introduced by Richard Wood under a different name (Wood 1976, 1978).…

Toward compact double categories: Part 2

Last time we framed the puzzle of axiomatizing categorical duality and introduced new double-categorical tools, culminating with the twisted Hom functor. We’ll now get straight to the point by proposing a definition of a compact double category. After that we’ll fill in a few remaining technical…

Toward compact double categories: Part 1

Lately I’ve been pursuing formal category theory using double categories. A first step in this program is to abstract from categories to category objects in a double category; for instance, internal categories are category objects in a double category of spans and enriched categories are category…

Algebras are promonads

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

The beginning of a series of posts answering the question “why double categories?”. Our first answer is that double categories give the algebra of relations from universal properties.

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.

Cartesian double theories

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

Unbiased monoidal categories are pseudo-elements

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

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…

Research opportunities at Topos in early 2023

We invite applications to join Topos Institute as a Research Associate (summer 2023) or as a Research Software Engineer (regular position).

Decorated cospans via the Grothendieck construction

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

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 \mathsf{Cat}. In fact, we suggest a general procedure for doing the Grothendieck construction on any structure…

Diagrammatic equations and multiphysics, Part 2

Previously, we surveyed some of the fundamental concepts from our paper on the use of diagrams to present equations from mathematical physics. In this blog post we take a look at how we can enrich our framework, introducing cartesian and symmetric monoidal products in order to be able to express…

Tim Hosgood, Evan Patterson

Diagrammatic equations and multiphysics, Part 1

Using diagrams to encode equations between physical quantities is something that has become more and more common over the past few decades. To a category theorist, however, the use of such diagrams is not as formal as one would perhaps like. In this new paper, we fix this problem: systematising…

Tim Hosgood, Evan Patterson

Call for 2022 Summer Research Associates

Come spend the summer at the Topos Institute! For early-career researchers, we’re excited to open up applications for our summer research associate program.

Topos seeks Research Software Engineer

We invite applications to join the Topos Institute as a Research Software Engineer, to work on software for compositional data integration and scientific modeling in the Julia programming language.