### Categorical Unification

#### Posted by David Corfield

A couple of postdoc positions have been advertised to work with Bob Coecke and Samson Abramsky on a research project – Categorical unification: where foundational physics, natural language and rational processes meet – in the Department of Computer Science in Oxford. In view of the Executive Summary, I’d be interested to see the full description of the project.

Recent advances show that category theory can be used to capture the essential behavioural properties of many complex systems, and provides the right language to study their foundational concepts across a broad range of disciplines, including the physical world, logical and deductive systems, the way that meaning is encoded into a sentence, and closely related, cognition.

Essential mathematical components of this work include monoidal categories, sheaf theory and coalgebra. The mathematical study of these is currently separate, but the common mathematical formalism underlying them suggests that they should be studied together, as a part of a whole. The ultimate aim is to develop a fully-integrated mathematical formalism for modelling the physical world, making deductions about it, and communicating those deductions linguistically – in short, a mathematical formalism for intelligent reasoning.

These developments would go hand-in-hand with ongoing projects which aim to automate reasoning about linguistic meaning and quantum processes, by exploiting the logical content carried by the graphical languages which describe these areas.

This project addresses both of the Big Questions of this call, by proposing new paradigms for knowledge representation, language and reasoning, as well as for the artificial implementation of these. The models of meaning moreover model the cognitive mind in a similar manner as we model abstract mathematical reasoning.

In particular, I wonder what the second paragraph is hinting at.

## Re: Categorical Unification

Maybe I can trigger a reaction from somebody who knows more by making a critical comment: given that I know a bit of the Coecke-Abramsky program – which is all great as far as it goes – I find these suggestions a bit strong.

A while back we had a speaker in our colloquium introducing a talk on related matters with advertizements of how this program provides a foundation for all of quantum mechanics. But the formalism really formalizes just quantum kinematics and just for systems with finitely many degrees of freedom. This is of course precisely the case that quantum information theory is concerned with, where it is all about shuffling a bunch of qbits around. But it excludes already simple things like the particle on the line and it is a long way from there to a foundation of quantum physics as a whole.

Not that the necessary generalization are not possible and have not been considered: Stephan Stolz and Peter Teichner have written out many of the details as a low-dimensional example in the context of their work of quantum field theories, where quantum mechanics is formalized as a monoidal functor on a category of Riemannian 1-dimensional cobordisms with values in a suitable monoidal category of topological vector spaces.

This has received less attention than it deserves. I would enjoy seeing the thrust of the Coecke-Abramsky school be expanded to this general context of quantum physics.