Re: Sigma and Products

[Michael Shulman <mshulman@ucsd.edu>]

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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