Day's reflection theorem in reverse

Day's reflection theorem in reverse

[Harry Gindi]

Dear Categories Mailing list,

Day's reflection theorem gives us a way to construct biclosed monoidal
structures on reflective subcategories of the presheaf category of a
pro-monoidal category.  Is there a converse 'trace' theorem, allowing
us to construct promonoidal categories on dense subcategories of
monoidal biclosed categories under reasonable conditions?

Is there a useful and usable condition to determine if, when given the
data of a monoidal biclosed category B and a full dense subcategory A
c B, the trace of B on A exists and is a promonoidal structure on A?
It seems that Day's 1974 paper uses something like this to prove lemma
3.1.1, but it seems to use the fact that the monoidal biclosed
category B is the functor category [A,V] in an essential way, but I
don't see exactly where this is used to show the trace is
pro-monoidal, especially since Day claims that it only relies on the
existence of V-cotensors in B, the fact that B is biclosed monoidal,
and the fact that A is dense and full in B (see proof of 3.1.1).  Is
the proof incorrect, or am I missing something?

I ask this because I am trying to find the precise point of error In
Ittay Weiss's thesis extending the Boardman-Vogt tensor product to
Dendroidal Sets.  If anyone knows the exact point in the proof where
it all goes wrong, I'd really like to know. If the strategory of proof
in 3.1.1 can be applied without using that B is [A,V], it seems like
Weiss's proof should have worked.

Thank you for your time and attention!

Your humble servant,

Harry J. Gindi


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]