Day tensors of fibered categories [009F]
I have been thinking about monoidal closed structures induced by slicing over a monoid, which has been considered by Combette and Munch-Maccagnoni as a potential denotational semantics of destructors à la C++. It occurred to me that this construction is, in fact, almost a degenerate case of Day convolution on an internal monoidal category — and this made me realize that there might be a nice way to understand Day convolution in the language of fibered categories. In fact, many of these results (in particular the relativization to internal monoidal categories) are probably a special case of Theorem 11.22 of Shulman’s paper on enriched indexed categories.
Under appropriate assumptions, we may also compute a “Day hom” by adjointness.
I believe, but did not check carefully, that when \(E\) and \(F\) are discrete fibrations over a semimonoidal category \({\mathopen {}\left (B,\otimes ,\alpha \right )\mathclose {}}\) then the Day tensor is precisely the discrete fibration corresponding to the (contravariant) Day convolution of the presheaves corresponding to \(E\) and \(F\). Likewise when \({\mathopen {}\left (B,\otimes ,I,\alpha ,\lambda ,\rho \right )\mathclose {}}\) is monoidal, it appears that the Day unit corresponds precisely to the traditional one.
There remain some interesting directions to explore. First of all, the claims above would obviously lead to a new construction of the Day convolution monoidal structure on the 1-category of discrete fibrations on \(B\) that coincides with the traditional one up to the Grothendieck construction. But in general, we should expect to exhibit both \({\mathbf {Cat}}_{/B}\) and \(\mathbf {Fib}_{B}\) as monoidal bicategories, a result that I have not seen before.
Conjecture [009E] is highly non-trivial, as monoidal bicategories are extremely difficult to construct explicitly. I am hoping that Mike Shulman’s ideas involving monoidal double categories could potentially help.