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Stream: learning: questions

Topic: Closed discrete fibration?

fosco (Nov 09 2022 at 21:21):

I have a discrete fibration $p : E\to B$ and $B$ is a monoidal closed category; I would like to know under which conditions $E$ is also monoidal closed (and $p$ 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.

fosco (Nov 09 2022 at 21:24):

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

The specific fibration $p : E\to B$ 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 $E$ in terms of properties of $p$, and hopefully make the counterexample I have more elegant.

Christian Williams (Nov 09 2022 at 22:27):

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

fosco (Nov 10 2022 at 08:22):

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

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.

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.