Slick proof that f (x) (g+h) = (f (x) g) + (f (x) h) in a monoidal category with 0, biproducts and duals

Slick proof that f (x) (g+h) = (f (x) g) + (f (x) h) in a monoidal category with 0, biproducts and duals

[Jamie Vicary]

Hi,

In a category with a zero object and biproducts we obtain a unique
enrichment in commutative monoids, which we will write as +. If the
category is also monoidal with left and right duals for objects, then
the tensor product distributes over +, in the sense that
    f (x) (g+h) = (f (x) g) + (f (x) h)
for all morphisms f,g,h with g and h in the same hom-set.

I have a proof of this but it is a bit clunky, and rather long. Can
anyone give a beautiful one?

Best wishes,
Jamie.


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Re: Slick proof that f (x) (g+h) = (f (x) g) + (f (x) h) in a monoidal category with 0, biproducts and duals

[Richard Garner]

Dear Jamie,

Here is one possible proof. The 2-category of categories with
biproducts and zero objects and functors preserving them is
biequivalent to the 2-category of CMon-enriched categories with finite
coproducts and arbitrary CMon-functors between them (because finite
coproducts are absolute in the CMon-enriched world).

If C is a category with biproducts, a zero object, and monoidal with
left and right duals, then it is in particular monoidal biclosed;
whence the tensor product A (x) (-) : C ----> C preserves colimits, so
in particular biproducts and the zero object. Via the above
biequivalence, it follows that A (x) (-) : C ---> C is a CMon-enriched
functor for each A. Thus given a map f : A --->B and objects X,Y in C,
we have mappings

C(X,Y) ---> C(A(x)X, A(x)Y) ---> C(A(x)X, B(x)Y)

in CMon, whose underlying mapping in Set sends g : X ---> Y to f (x) g
: A(x)X --> B(x)Y. That these mappings lift to CMon expresses
precisely the distributivity you state above.

Best,
Richard


On 11 October 2013 23:24, Jamie Vicary <jamievicary@gmail.com> wrote:
> Hi,
>
> In a category with a zero object and biproducts we obtain a unique
> enrichment in commutative monoids, which we will write as +. If the
> category is also monoidal with left and right duals for objects, then
> the tensor product distributes over +, in the sense that
>     f (x) (g+h) = (f (x) g) + (f (x) h)
> for all morphisms f,g,h with g and h in the same hom-set.
>
> I have a proof of this but it is a bit clunky, and rather long. Can
> anyone give a beautiful one?
>
> Best wishes,
> Jamie.


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