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Aaron David Fairbanks (Jan 23 2025 at 23:03):A few years ago, David Spivak wrote a paper called Functorial Aggregation, which was available on the arXiv. Now a new version by David, Richard Garner, and me is getting published! It's changed quite a lot from the older version.
The paper looks at various structures in the category of "single-variable polynomials": functors of the form (equivalently, parametric right adjoint functors ).
Something really neat and surprising, if you haven't seen it before, is Ahman and Uustalu's result that comonoids in (p.r.a. comonads on ) are categories.
However, it gets more interesting. We can consider "multi-variable polynomials" (p.r.a. functors for sets and ); and even more generally we can consider "polynomials with variables and values indexed by categories" (p.r.a. functors for categories and ). Extending Ahman and Uustalu's result, Richard pointed out in this video from 2019 that the framed bicategory of comonoids in (where 1-cells are bicomodules) is exactly the bicategory of these category-variable category-valued polynomials. A proof is given in the paper.
Not only the concept of a category but also the appropriate generalized concept of multi-variable polynomial fall right out of the concept of single-variable polynomial!
Kevin Carlson (Jan 24 2025 at 00:32):Congrats, looks great!
Jean-Baptiste Vienney (Jan 24 2025 at 01:28):Congrats @Aaron David Fairbanks !!