An Introduction to Topos Physics

TL;DR

This paper introduces topos theory as a mathematical framework to reformulate quantum physics, aiming to create a more classical-like, neo-realistic understanding of quantum phenomena using category theory and formal languages.

Contribution

It simplifies and explains the topos approach to quantum physics, making the complex ideas accessible and providing a foundation for representing physical theories within a topos.

Findings

Topos theory offers a new formal language for quantum physics.

The approach enables neo-realistic interpretations of quantum phenomena.

Simplified exposition makes the topos approach accessible to non-experts.

Abstract

The basic notion of how topoi can be utilized in physics is presented here. Topos and category theory serve as valuable tools which extend our ordinary set-theoretical conceptions, can further the study of quantum logic and give rise to new and 'neo-realistic' descriptions of quantum physics, i.e. make possible the construction of a general scheme for quantum physics, which 'looks like' the classical one. A novel way of writing theories of physics, by representing a formal language of a physical system, into an appropriate topos, is analyzed here. This novel approach is initiated by C.J. Isham and A. Doering and the major part of this paper comes from there; however there has been an effort in simplifying the basic ideas and do this work accessible to 'less experts' of the field. A minor, but not less important, part of this paper is devoted to resuming, in a concise way the background knowledge of logic, quantum logic and category theory, that is necessary for anyone who wishes to follow the whole topos approach.