Representations of 2-groups are nice examples of symmetric monoidal bicategories with non-trivial coherence data (the "15j" symbols which categorify the "6j symbols" from representation categories of groups). See eg. "2-group representations for spin foams", by Baratin and Wise for a pointer to further literature, http://arxiv.org/abs/0910.1542. Regards, Bruce Bartlett On Mon, Jul 16, 2012 at 6:49 AM, Steve Lack <steve.lack@mq.edu.au> wrote:
On 15/07/2012, at 4:11 AM, Mike Stay wrote:
On Thu, Jul 12, 2012 at 7:59 AM, Roman Krenický <roman.krenicky@cs.manchester.ac.uk> wrote:
Dear all,
I'm looking for examples of symmetric monoidal bicategories (where the structure is genuinely weak, i.e. the various isomorphisms are not identities) and I would appreciate some help. -- Actually, strict 2-categories with (genuinely) weak monoidal structure would be even more interesting, but I found it almost impossible to find anything on that.
As I am using these categories as models, I need some structure that is "concrete enough" to do calculations with (while being as simple as possible).
Right now I'm considering the bicategory of rings (or monoids or fields), bimodules over them, and bimodule homomorphisms, where the monoidal structure is defined by the tensor product etc. (Pointers to detailed accounts of this category would be very much appreciated, too. I've only found fairly sketchy mentions in the literature.)
I would be grateful about any other examples of this kind!
Spans of sets. The cartesian product of sets has an associator, so the tensor product of spans does, too.
That's a good example; it's also very similar to the example of modules.
For any braided monoidal category V with coequalizers of reflexive pairs, which are preserved by tensoring on either side, there's a monoidal bicategory Mod(V) in which the objects are the monoids in V, the 1-cells from a monoid M to a monoid N are objects equipped with a left M-action and a right N-action satisfying the obvious compatibility condition, and the 2-cells are the morphisms in M compatible with the two actions.
If you start with the monoidal category Ab of abelian groups, with the usual tensor product, you get the bicategory of rings, bimdules, and bimodule homomorphisms.
If you start with the *opposite category* of Set, with tensor product given by the cartesian product in Set (and so by the coproduct in Set^op), then a monoid is just a set, an object with compatible left and right actions is a Span, and a 2-cell is a morpihsm of spans. It then turns out that Mod(Set^op) is Span^co, where the co means that the direction of the 2-cells is reversed.
You could also consider a larger monoidal bicategory Mod-V, where the objects are not just monoids in V but V-enriched categories; then the 1-cells will be enriched profunctors.
Steve Lack.
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