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Stream: deprecated: our papers

Topic: category of real numbers: the addition as a biproduct

dusko (Jan 29 2021 at 07:09):

i wrote this paper
https://arxiv.org/abs/2007.10057
as a chapter for a book intended for logicians. but to make a point that i needed to make there, in Sec.6 i worked out a category where the objects are real numbers, and the morphisms are polarized simulations. the crazy thing, which i think might deserve attention of category theorists, is that the addition of real numbers turns out to be the BIPRODUCT in this category.

so what? well, this category basically discovers linear algebra as kan extensions.

take a profunctor $m\times n \to R$, where $m$ and $n$ are discrete finite categories, a.k.a. finite sets. such profunctors are commonly called matrices. each of them as a profunctor induces an adjunction between the categories $R^m$ and $R^n$. since $m$ and $n$ are discrete, the kan extensions are induced by the biproducts. the two linear operators induced by a matrix are the two kan extensions of the profunctor.

if you know Conway's On Numbers and Games, then Sec.6 can be read on its own, i think. else if you liked Samson Abramsky's interaction categories, then the rest of the paper might be of some interest.

either way, i thought that discovering linear operators as kan extensions (left=right) was nice.