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

Joachim Kock

A concrete computation --- twelve slidings with sixteen tiles --- reveals that certain commutativity phenomena occur in every double semi\-group. 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.

Journal of Homotopy and Related Structures, Vol. 2(2007), No. 2, pp. 217-228