Re: Examples of symmetric monoidal bicategories

[Bruce Bartlett <brucehbartlett@gmail.com>]

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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