John Baez wrote, concerning categories whose morphisms look like this: * * * \ / / | /\ \ / | \/ | * *
First of all, Jules Bean leaves it quite open-ended exactly which category he is talking about. He is actually talking about a large number of interesting categories each with their own description. Secondly, the usual definition of bimonoid involves structures and laws that are not so natural from the topological viewpoint - i.e., certain morphisms are decreed to be equal even when their corresponding embedded graphs are not isotopic. Whether this is good or bad depends on what you're trying to do.
Agreed on all counts. It's also interesting (to me) to consider the dual diagrams, which look something like computads (depending on exactly which version of the category above you're using); this gives different geometric intuitions. There's more about this, and higher-dimensional generalizations, in Ross Street, Categorical structures, in Handbook of Algebra I, ed. M. Hazewinkel, North-Holland, 1996, pp. 529-577. Tom 26-Sep-2001 13:10:01 -0300,1644;000000000001-00000025