Hi Neil, In my paper "Framed bicategories and monoidal fibrations" I studied the more general situation of a fibration between monoidal categories, which is also a monoidal functor in a compatible way. When B (but not necesarily E) is cartesian monoidal, this situation is equivalent to saying that each fiber category is monoidal in a compatible way. If E is regarded as a "B-indexed category" and Sigma as an "indexed coproduct," then the condition you mention (when phrased more precisely as "\times (or \otimes) preserves opcartesian arrows in E") is one way of saying that the "tensor product of E preserves indexed coproducts in each variable." Another way of saying this same thing, which is equivalent when the left adjoints Sigma satisfy the Beck-Chevalley condition for pullback squares in B, is that the "Frobenius maps" Sigma_f ( P \otimes f^* Q) --> (Sigma_f P) \otimes Q are isomorphisms, where in this case \otimes represents the monoidal structure in a single fiber category. (I didn't state this equivalence explicitly in the paper, but it is the dual of 13.15.) Finally, just as in the unindexed case, both conditions follow automatically if the monoidal structure of E is "closed" in a suitable sense. One version of this is that if each fiber is closed monoidal and so are the reindexing functors, then the Frobenius condition is automatic (this is a well-known property of adjunctions between closed monoidal categories), and thus (assuming the Beck-Chevalley condition) the property you ask about also follows. Best, Mike On Fri, Dec 10, 2010 at 1:45 AM, Neil Ghani <Neil.Ghani@cis.strath.ac.uk> wrote:
Hi
Sorry ... nothing deep or fundamental here.
Here is a simple question which someone may know the answer to.
Lets say we have a fibration p:E -> B with left adjoints Sigma_f for every reindexing functor f^*. Lets say further that E has products and p preserves them.
Are there simple conditions under which we have
Sigma_{f \times g} (P \times Q)
iso
(Sigma_f P) \times (Sigma_g Q)
All the best Neil
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