Note on commutativity in double semigroups and two-fold monoidal categories

TL;DR

This paper investigates commutativity in double semigroups and two-fold monoidal categories, revealing that certain structures inherently exhibit commutative properties, impacting the understanding of higher categorical and homotopical structures.

Contribution

It demonstrates that all cancellative and inverse double semigroups are commutative and shows that strictly associative two-fold monoidal categories are degenerate symmetric.

Findings

Cancellative double semigroups are commutative.

Inverse double semigroups are commutative.

Strictly associative two-fold monoidal categories are degenerate symmetric.

Abstract

A concrete computation -- twelve slidings with sixteen tiles -- reveals that certain commutativity phenomena occur in every double semigroup. This can be seen as a sort of Eckmann-Hilton argument, but it does not use units. The result implies in particular that all cancellative double semigroups and all inverse double semigroups are commutative. Stepping up one dimension, the result is used to prove that all strictly associative two-fold monoidal categories (with weak units) are degenerate symmetric. In particular, strictly associative one-object, one-arrow 3-groupoids (with weak units) cannot realise all simply-connected homotopy 3-types.