# Space in Monoidal Categories

## TL;DR

This paper explores the structure of monoidal categories in quantum theory, interpreting spacetime and quantum information processes through categorical constructs like idempotent subunits and restriction operations.

## Contribution

It introduces a novel categorical framework for modeling spacetime and quantum information, linking algebraic localization with spacetime propagation within monoidal categories.

## Key findings

- Quantum teleportation depends on the intersection of causal futures.
- Idempotent subunits form a meet-semilattice in braided monoidal categories.
- Restriction operations model spacetime propagation internally.

## Abstract

The category of Hilbert modules may be interpreted as a naive quantum field theory over a base space. Open subsets of the base space are recovered as idempotent subunits, which form a meet-semilattice in any firm braided monoidal category. There is an operation of restriction to an idempotent subunit: it is a graded monad on the category, and has the universal property of algebraic localisation. Spacetime structure on the base space induces a closure operator on the idempotent subunits. Restriction is then interpreted as spacetime propagation. This lets us study relativistic quantum information theory using methods entirely internal to monoidal categories. As a proof of concept, we show that quantum teleportation is only successfully supported on the intersection of Alice and Bob's causal future.