arXiv subjects

Aaron David Fairbanks

Publications and source records attributed to Aaron David Fairbanks.

5 recordsLinked to original sources

arXiv2026

Comonads as spaces

Comonads on Set generalize both categories and topological spaces. Expanding upon Garner's work on ionads, we develop aspects of the theory of topological spaces for arbitrary comonads on arbitrary categories. Our approach is centered around density comonads, which provide an abstraction of subbases. We study subbases as well as bases in terms of density comonads, and we study continuous maps of comonads in terms of functors between coalgebra categories, with definitions that recover the usual notions for topological spaces. Whereas Ahman and Uustalu characterized categories as precisely the polynomial comonads on Set, we characterize topological spaces as precisely the density comonads of diagrams of subsets of a set, which are familiar as topological subbases. We show that every comonad on Set has an underlying topological space, and that this construction is a reflection with respect to continuous maps; similarly, every comonad on Set has an underlying small category, and this construction is a coreflection. We also show that the category of all comonads on Set with continuous maps is complete, and that its full subcategory of accessible comonads is cocomplete. Continuous maps and ordinary comonad morphisms form a double category, which, in the case of the polynomial comonads on Set, recovers the double category of functors and retrofunctors of Clarke and Di Meglio. We find topological intuition for these concepts in terms of "halos", an abstraction of infinitesimal neighborhoods of points, defined as formal limits of neighborhood systems. We include a long appendix of counterexamples, many applicable to general (co)monad theory rather than the particular concerns of this text.

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arXiv2026

Monads in 2-categories

This is a condensed overview of the formal theory of monads in a 2-category. Commutative diagrams and string diagrams are given side by side. In the string diagrams, each concept is visualized in a way that suggests its properties using topological intuition. For example, monads themselves appear as channels of fluid, suggesting the unit and associativity laws by topological deformation. We cover monads, modules, (co)lax 1-cells and the 2-cells between them, algebra objects, bimodules, distributive laws, codensity monads, and pushforward monads. In addition to the standard 2-categories of monads in a 2-category, we also define two double categories of monads in a 2-category. For example, applied to spans, this yields the two double categories of categories, functors, and retrofunctors upon restricting to the 1-cells carried by functions.

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arXiv2026

Doubly weak double categories

We propose a definition of double categories whose composition of 1-cells is weak in both directions. Namely, a doubly weak double category is a double computad -- a structure with 2-cells of all possible double-categorical shapes -- equipped with all possible composition operations, coherently. We also characterize them using "implicit" double categories, which are double computads having all possible compositions of 2-cells, but no compositions of 1-cells; doubly weak double categories are then obtained by a simple representability criterion. Finally, they can also be defined by adding a "tidiness" condition to the double bicategories of Verity, or to the cubical bicategories of Garner.

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arXiv2025

On Traces in Categories of Contractions

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).

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arXiv2025

Functorial aggregation

We study polynomial comonads and polynomial bicomodules. Polynomial comonads amount to categories. Polynomial bicomodules between categories amount to parametric right adjoint functors between corresponding copresheaf categories. These may themselves be understood as generalized polynomial functors. They are also called data migration functors because of applications in categorical database theory. We investigate several universal constructions in the framed bicategory of categories, retrofunctors, and parametric right adjoints. We then use the theory we develop to model database aggregation alongside querying, all within this rich ecosystem.

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