Alex- The reference you are looking for is Cartesian Bicategories II by Carboni, Kelly, Walters, and Wood. Here is a link to the .pdf on the TAC homepage. http://www.tac.mta.ca/tac/volumes/19/6/19-06.pdf Nick Alex Hoffnung said the following on 23/11/2010 00:44:
Dear List,
The weakest form of limits in a bicategory are defined by equivalences of hom-categories in place of the slightly more strict version defined by isomorphisms. Then the process of defining a monoidal structure on a bicategory with finite products takes on a slightly different flavor in each case.
In the weak case, given a pair of 1-cells one must *choose* a monoidal product, whereas in the latter case, a *unique choice* of monoidal product can be obtained from the one-dimensional aspect of the universal property.
I would guess that in the former case, one should not worry too much about how to choose the product 1-cells, since the universal property will come to the rescue when checking that one has indeed defined a monoidal structure on the bicategory.
Does anyone know of a reference which proves something along these lines?
Best, Alex
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