## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://www.springer.com/mathematics/geometry/journal/40062.
**

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