Category theory with examples in probability theory

TL;DR

This paper introduces fundamental category theory concepts with illustrative examples from probability theory, highlighting their applications and challenges in formalizing probability spaces within categorical frameworks.

Contribution

It develops basic category theory concepts and demonstrates their application to probability theory, emphasizing the integration of measurement and probabilistic tools.

Findings

Examples of categories, functors, and natural transformations in probability context

Discussion of Lawvere and Giry approaches to categorical probability

Highlighting challenges in defining morphisms between probability spaces

Abstract

The basic concepts of category theory are developed and examples of them are presented to illustrate them using measurement theory and probability theory tools. Motivated by Perrone's workarXiv:1912.10642 where notes on category theory are developed with examples of basic mathematics, we present the concepts of category, functor, natural transformation, and products with examples in the probabilistic context. The most prominent examples of the application of Category Theory to Probability Theory are the Lawvere (available at ncatlab.org/nlab/files/lawvereprobability1962.pdf.) and Giry (avaible at https://doi.org/10.1007/BFb0092872) approaches. However, there are few categories with objects as probability spaces due to the difficulty of finding an appropriate condition to define arrows between them