The n-Category Café

Thanks! It’s exciting to finally finish something again; it feels like forever since I did that. (-:

clearly the above list of examples wants to contain also monoidal derivators?

Yes, but. A monoidal derivator is, indeed, an indexed monoidal category, and its reindexing functors indeed have left adjoints. However, it fails to satisfy all the Beck-Chevalley conditions that we need for this paper. A derivator, by definition, satisfies the Beck-Chevalley condition for comma squares, but not all the pullback squares we need are comma squares in $Cat$.

If we restrict the domain categories of our derivator to groupoids, of course, we do get all the conditions necessary, since for groupoids, comma categories and (pseudo) pullbacks coincide. It seems that the way of thinking presented in this paper really only applies to indexings by “groupoidal” objects.

However, you can still do many of the same things with a monoidal derivator. In particular, the essential construction of the current paper is the same as that of this old paper: from an indexed monoidal category we build a bicategory with $Hom(A,B) = C^{A\times B}$. (We then get refined notions of trace from Kate’s bicategorical traces in this bicategory.)

But clearly, if $D$ is a monoidal derivator, then the bicategory you want to build is a bicategory of profunctors, with $Hom(A,B) = D(B^{op}\times A)$. Traces in this bicategory exhibit many of the same phenomena as in the bicategory constructed from a monoidal fibration, and I think we’ll have some other very interesting things to say about them in the near future. Stay tuned!

Interesting question! What we have in mind, insofar as we have anything in mind beyond the schematic pictures you see, are surfaces with singularities (these being the “standard” surface diagrams for monoidal bicategories). For instance, figure 15(b) can be seen as a simple topological deformation between two such surfaces. But maybe this is the same thing; I don’t know what a “foam” is.

Do your findings suggest anything interesting in the case of predicate logic and in general example 3.9 on hyperdoctrines?

Hyperdoctrines are very interesting indexed monoidal categories. As you know, our string diagram calculus came from them. But I don’t think that duality and trace are going to have much to say in hyperdoctrines. A hyperdoctrine is cartesian monoidal, so there are no interesting dualities, and even if you looked at some non-cartesian “quantum hyperdoctrine”, traces in a poset aren’t going to be very interesting.

What about group or groupoid representations? We spoke about these in the context of the Beck-Chevalley condition back here.

Well, groupoid (hence also group) representations are an indexed monoidal category, as I mentioned above. And we found the same thing that Jeffrey Morton did – the Beck-Chevalley condition for groupoids (including groups, as a special case) only holds for pseudo pullbacks, not strict ones in general. We have a concrete counterexample, too – the strict pullback square $$\array{ G & \to & G \\ \downarrow & & \downarrow \\ G & \to & G\times G }$$ is almost never going to satisfy the Beck-Chevalley condition. In fact, the failure of this Beck-Chevalley condition is crucial for the interesting fixed-point theory applications, since it means that the “shadow” of a groupoid is different from its homotopy quotient, and the shadow is where the refined fixed-point invariants like the Reidemeister trace live.

To reply to a few points raised at the discussion you linked to, which the participants probably know by now:

• When looking for a Beck-Chevalley condition, there’s always a map in one direction; the question is precisely whether it is an isomorphism.
• Any strict pullback of a fibration between groupoids is also (up to equivalence) a pseudo pullback. And any surjection of groups is a fibration between one-object groupoids. So the Beck-Chevalley condition holds for strict pullbacks of groups where one leg is a surjection.
• There is a Beck-Chevalley condition for squares of categories, but it requires the square to be a comma square, not a pseudo pullback – it just so happens that the two coincide for groupoids.

Learning Peirce’s system, you spend a lot of time following his rules about ‘seps’, which designate negation. Does this correspond to a feature of the particular kind of hyperdoctrine that is classical logic that one might find in some more general indexed monoidal categories?

As I mentioned, this diagram of Osman is very Peircean.