# Categorical Semantics for Time Travel

## TL;DR

This paper develops a categorical framework to model quantum theories with closed timelike curves, capturing existing models and ensuring relativistic causality constraints are respected.

## Contribution

It introduces a weakened traced symmetric monoidal category framework that unifies and describes quantum CTC models, including the first compositional D-CTC model.

## Key findings

- The framework captures P-CTC and D-CTC models.
- Provides a compositional description of D-CTC.
- Ensures causality constraints are maintained in the models.

## Abstract

We introduce a general categorical framework to reason about quantum theory and other process theories living in spacetimes where Closed Timelike Curves (CTCs) are available, allowing resources to travel back in time and provide computational speedups. Our framework is based on a weakening of the definition of traced symmetric monoidal categories, obtained by dropping the yanking axiom and the requirement that the trace be defined on all morphisms. We show that the two leading models for quantum theory with closed timelike curves---namely the P-CTC model of Lloyd et al. and the D-CTC model of Deutsch---are captured by our framework, and in doing so we provide the first compositional description of the D-CTC model. Our description of the D-CTC model results in a process theory which respects the constraints of relativistic causality: this is in direct contrast to the P-CTC model, where CTCs are implemented by a trace and allow post-selection to be performed deterministically.