Physics, Topology, Logic and Computation: A Rosetta Stone

TL;DR

This paper explores deep analogies between physics, topology, logic, and computation, using category theory to unify these fields and facilitate understanding without prior specialized knowledge.

Contribution

It provides a clear, accessible explanation of how closed symmetric monoidal categories connect concepts across physics, topology, logic, and computation.

Findings

Analogies between quantum physics and topology clarified

Category theory unifies diverse concepts across disciplines

Accessible exposition without prior specialized knowledge

Abstract

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much like a "cobordism". Similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of "closed symmetric monoidal category". We assume no prior knowledge of category theory, proof theory or computer science.