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Stream: learning: questions

Topic: Closed discrete fibration?


view this post on Zulip fosco (Nov 09 2022 at 21:21):

I have a discrete fibration p:EBp : E\to B and BB is a monoidal closed category; I would like to know under which conditions EE is also monoidal closed (and pp is strong monoidal). Even before that I would like to know if this is a natural request, or if I should not expect something like that to be easily true.

Discreteness of the fibration is what prevents me to follow the usual yoga: a fibration is such and such if every fiber is such and such, and reindexings are homomorphisms of such and such. Now there is no monoidal closed structure on the fibers, only on the whole category.

view this post on Zulip fosco (Nov 09 2022 at 21:24):

If it's relatively easy to understand under which conditions pp is a strong monoidal functor between monoidal categories (something on the lines of "the associated presheaf Fp:BSetF_p : B\to Set is a monoid under Day convolution", iirc) closedness seems to be a more elusive concept.

The specific fibration p:EBp : E\to B I have to work with is quite likely not closed in this sense, but I would like to see a necessary condition for closure of EE in terms of properties of pp, and hopefully make the counterexample I have more elegant.

view this post on Zulip Christian Williams (Nov 09 2022 at 22:27):

couldn't you still have p:BopClMonCatp^\ast :B^{op}\to \mathrm{ClMonCat}, just landing in discrete categories? so the fibers would be cancellative monoids

view this post on Zulip fosco (Nov 10 2022 at 08:22):

Is it enough that the fibration pp has each fiber a (cancellative) monoid in order for this to be true? Or some other condition is necessary?

view this post on Zulip John Baez (Nov 10 2022 at 10:10):

I forget if Christina and Joe studied discrete monoidal fibrations in their paper. It'd be good to ask them.

view this post on Zulip Joe Moeller (Nov 10 2022 at 15:25):

We didn't bring up discreteness, but of course everything we did applies to that case. The thing we didn't consider was closedness. It did cross our minds, but the most obvious route didn't seem right, so we didn't pursue it.