Functorial aggregation

by   David I. Spivak, et al.

Aggregating data in a database could also be called "integrating along fibers": given functions π E→ D and s E→ R, where (R,⊛) is a commutative monoid, we want a new function (⊛ s)_π that sends each d∈ D to the "sum" of all s(e) for which π(e)=d. The operation lives alongside querying – or more generally data migration – in typical database usage: one wants to know how much Canadians spent on cell phones in 2021, for example, and such requests typically require both aggregation and querying. But whereas querying has an elegant category-theoretic treatment in terms of parametric right adjoints between copresheaf categories, a categorical formulation of aggregation – especially one that lives alongside that for querying – appears to be completely absent from the literature. In this paper we show how both querying and aggregation fit into the "polynomial ecosystem". Starting with the category 𝐏𝐨𝐥𝐲 of polynomial functors in one variable, we review the relatively recent results of Ahman-Uustalu and Garner, which showed that the framed bicategory ℂ𝐚𝐭^♯ of comonads in 𝐏𝐨𝐥𝐲 is precisely the right setting for data migration: its objects are categories and its bicomodules are parametric right adjoints between their copresheaf categories. We then develop a great deal of theory, compressed for space reasons, including local monoidal closed structures, a coclosure to bicomodule composition, and an understanding of adjoints in ℂ𝐚𝐭^♯. Doing so allows us to derive interesting mathematical results, e.g. that the ordinary operation of transposing a span can be decomposed into the composite of two more primitive operations, and then finally to explain how aggregation arises, alongside querying, in ℂ𝐚𝐭^♯.



There are no comments yet.


page 1

page 2

page 3

page 4


Quotients of span categories that are allegories and the representation of regular categories

We consider the ordinary category Span(C) of (isomorphism classes of) sp...

Why FHilb is Not an Interesting (Co)Differential Category

Differential categories provide an axiomatization of the basics of diffe...

Categories for Me, and You?

A non-self-contained gathering of notes on category theory, including th...

A 2-Categorical Study of Graded and Indexed Monads

In the study of computational effects, it is important to consider the n...

Pattern formation in a cell migration model with aggregation and diffusion

In this paper, we study pattern formations in an aggregation and diffusi...

Hypergraph Categories

Hypergraph categories have been rediscovered at least five times, under ...

Self-reporting and screening: Data with current-status and censored observations

We consider survival data that combine three types of observations: unce...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.