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I have a discrete fibration and is a monoidal closed category; I would like to know under which conditions is also monoidal closed (and 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.
If it's relatively easy to understand under which conditions is a strong monoidal functor between monoidal categories (something on the lines of "the associated presheaf is a monoid under Day convolution", iirc) closedness seems to be a more elusive concept.
The specific fibration 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 in terms of properties of , and hopefully make the counterexample I have more elegant.
couldn't you still have , just landing in discrete categories? so the fibers would be cancellative monoids
Is it enough that the fibration has each fiber a (cancellative) monoid in order for this to be true? Or some other condition is necessary?
I forget if Christina and Joe studied discrete monoidal fibrations in their paper. It'd be good to ask them.
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.