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:
