The Word Problem for Braided Monoidal Categories is Unknot-Hard

TL;DR

This paper proves that determining equivalence in braided monoidal categories is computationally as hard as unknotting in knot theory, highlighting significant complexity in categorical algebra.

Contribution

It establishes the unknot-hardness of the word problem for braided monoidal categories and extends this result to Gray categories, proposing a conjecture on their decidability.

Findings

Word problem for braided monoidal categories is at least as hard as unknotting.

The result extends to Gray categories.

Conjecture that the word problem for Gray categories might be decidable.

Abstract

We show that the word problem for braided monoidal categories is at least as hard as the unknotting problem. As a corollary, so is the word problem for Gray categories. We conjecture that the word problem for Gray categories is decidable.