Mathematics – Category Theory
Scientific paper
2006-08-17
Mac Lane Memorial Volume, J. Homotopy Rel. Struct., 2 (2007), 217-228.
Mathematics
Category Theory
10 pages, LaTeX, uses Paul Taylor's diagrams package. An animated graphical version of the computation mentioned in the abstra
Scientific paper
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.
Kock Joachim
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