Does every enriched functor preserve tensors?

Let P be a k-linear semisimple abelian rigid monoidal category with finite dimensional (over k) Hom-spaces (for a field k).

By a tensored P-category we mean a P-category which admits tensors with objects in P, i.e. for every two objects X,Y in the category and P in P, there is an object PX with an isomorphism:
[PX,Y][P,[X,Y]].

Does every enriched functor F between tensored P-categories preserve tensors? (i.e. the natural induced map PF(X)F(PX) is isomorphism) What happens if the categories are also abelian and F is exact?

Answer

No. The functor “take a vector space to its double dual” is linear but does not preserve tensors with infinite-dimensional vector spaces.

Enriched functors are automatically “lax tensored”. An enriched functor F provides a natural map [X,Y]F[FX,FY] for any X,Y. Consider setting Y=PX, and study:
id[PX,PX][P,[X,PX]]F[P,[FX,F(PX)]][PFX,F(PX)].
This gives a canonical map PFXF(PX) which is natural in P,X and compatible with associativity an unit data. It just isn’t automatically an isomorphism.


On first reading, I missed the requirement that P be rigid. In that case the answer is Yes, enriched functors are automatically tensored. In general, a-priori-lax constructions become strong when there is enough dualizability. Consider the composition
F(PX)PPF(PX)PF(PPX)PF(X)
where the first map is the unit between P,P, the second is the lax monoidality constructed above, and the third is F of the counit. This should give an inverse to the lax monoidality.

Attribution
Source : Link , Question Author : Mostafa , Answer Author : Theo Johnson-Freyd

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