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

By Joachim Kock Download
Mac Lane Memorial Volume, J. Homotopy Rel. Struct., 2 (2007), 217-228.

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.

The commutativity phenomenon is the following.

Proposition. For any sixteen elements a,b,... in any double semigroup, this equation holds:

(The empty boxes represent fourteen nameless elements, the same on each side of the equation, and in the same order.)

Proof. The proof consists of twelve slidings, each representing a strict equality.
Press the button to play the proof:

Last updated: 2008-07-13 by Joachim Kock.