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Stream: community: our work

Topic: Aaron David Fairbanks

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 $\mathbf{Poly}$ of "single-variable polynomials": functors $\mathbf{Set} \to \mathbf{Set}$ of the form $X \mapsto \sum_{i \in I} X^{S_i}$ (equivalently, parametric right adjoint functors $\mathbf{Set} \to \mathbf{Set}$).

Something really neat and surprising, if you haven't seen it before, is Ahman and Uustalu's result that comonoids in $\mathbf{Poly}$ (p.r.a. comonads on $\mathbf{Set}$) are categories.

However, it gets more interesting. We can consider "multi-variable polynomials" (p.r.a. functors $\mathbf{Set}^A \to \mathbf{Set}^B$ for sets $A$ and $B$); and even more generally we can consider "polynomials with variables and values indexed by categories" (p.r.a. functors $\mathbf{Set}^C \to \mathbf{Set}^D$ for categories $C$ and $D$). Extending Ahman and Uustalu's result, Richard pointed out in this video from 2019 that the framed bicategory of comonoids in $\mathbf{Poly}$ (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 !!

Aaron David Fairbanks (Feb 24 2025 at 04:59):

A new paper, On Traces in Categories of Contractions by Peter Selinger and me appeared on the arXiv today! This version is twelve pages long, with some extra stuff in appendices. We plan to write a longer version later.

Here is the abstract:

Traced monoidal categories are used to model processes that can feed their outputs back to their own inputs, abstracting iteration. The category of finite dimensional Hilbert spaces with the direct sum tensor is not traced. But surprisingly, in 2014, Bartha showed that the monoidal subcategory of isometries is traced. The same holds for coisometries, unitary maps, and contractions. This suggests the possibility of feeding outputs of quantum processes back to their own inputs, analogous to iteration. In this paper, we show that Bartha's result is not specifically tied to Hilbert spaces, but works in any dagger additive category with Moore-Penrose pseudoinverses (a natural dagger-categorical generalization of inverses).

Moore-Penrose pseudoinverses are really interesting. They can be defined in any dagger category, and just like the definition of inverse, the definition of pseudoinverse is purely equational. When the pseudoinverse of an arrow exists, it is unique. They also arise very naturally in relation to idempotents -- an arrow $f$ is pseudoinvertible if and only if it constitutes some isomorphism of dagger-idempotents $f \colon p \to q$ (where $p$ is basically projection onto the coimage of $f$ and $q$ is projection onto the image of $f$). Robin Cockett and @JS PL (he/him) wrote a very nice paper about pseudoinverses in 2023.

With pseudoinverses, one gets a generalized abstraction of "singular value decomposition" of arrows. This is what lets us prove the main result of our paper.