Toby St Clere Smithe


2023-12-19: My thesis is now in the Oxford University Research Archive, at DOI 10.5287/ora-kzjqyop2d.

2023-11-16: Animating categories: rendering morphisms dynamical with systems theory.

2023-09-16: Open factor graphs via decorated cospans.

2023-09-08: Passed my (long time coming) DPhil viva! Slides here.


What I do

I study the mathematical structures that give systems life, with a particular focus on the theory of active inference, a story about how cognitive systems work together and learn to navigate their worlds. I seek to bring this life to artificial systems that currently lack it: to build smarter computer assistants for science, tools to harness the power of biology, and ultimately more ecological societies.

Follow this link to read more about my current research.

Where I do it

Currently, I am a member of the VERSES Research Lab and the Topos Institute.

I am also available for consultancy work applying compositional methods to computational or intelligent systems.


I wrote my DPhil (PhD) at the University of Oxford, where I studied the compositional structure of cognitive systems with a focus on predictive coding, an account of how the brain might instantiate active inference. I proved that Bayesian updates compose optically, demonstrated that the chain rule of the relative entropy may be classified as a section of a fibration of statistical games, defined new categories of open dynamical systems, and thus showed how predictive coding emerges from an effort to give functorial semantics to certain ‘autoencoder’ games.

Before that, I completed a double MSc in Complex Systems Science at the École Polytechnique and École Normale Supérieure in Paris, and Chalmers University in Gothenburg, and I retain an interest in applying category-theoretic tools to the study of complex and adaptive systems.

I also hold a BSc in mathematics from the Open University and a BA in psychology, philosophy and physiology from the University of Oxford.