On braidings, syllepses, and symmetries
Cahiers Topologie
Géom. Différentielle Catég.
XLI (2000), 2-74
Abstract:
Starting from the
tensor product of Gray-categories, I define
4-dimensional teisi, which generalize Gray-categories, and derive some
indications for a hypothetical generalization to higher dimensions. The first
result is that 4-dimensional teisi with one object and one arrow are (semistrict)
braided monoidal 2-categories with trivial R~'s. Next, I combine the
idea that sylleptic 2-categories should be 5-dimensional teisi with one
object, one arrow and one 2-arrow with the above indications. The second
result is that this gives a notion of syllepsis which is equivalent to Day and
Street's. Similarly, symmetric 2-categories should be 6-dimensional teisi
with one object, one arrow, one 2-arrow and one 3-arrow, and the third
result is that this gives a notion of symmetry which is equivalent to Day and
Street's. These last two results are easily extended to somewhat weaker braided
and sylleptic monoidal 2-categories.
``The author acknowledges the support of the
Australian Research Council''
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Sjoerd E. Crans
School of Mathematics, Physics, Computing & Electronics
Macquarie University
NSW
Australia