Computability and Complexity of Categorical Structures

TL;DR

This paper explores the computational power of category theory, demonstrating that categorical structures can simulate Turing machines and solve problems like the Halting Problem, surpassing traditional computation models.

Contribution

It establishes that category theory can perform all Turing machine computations and solve problems in the arithmetic hierarchy, showing its superior computational capabilities.

Findings

Categories can simulate Turing machines

Categories can solve the Halting Problem

Category theory exceeds Turing machine power

Abstract

We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by categories. Since there are infinitary constructions in category theory, it is shown that category theory is strictly more powerful than Turing machines. In particular, categories can solve the Halting Problem for Turing machines. We also show that categories can solve any problem in the arithmetic hierarchy.