Richard Garner

David I. Spivak, Richard Garner, Aaron David Fairbanks

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.

Richard Garner, Alyssa Renata, Nicolas Wu

We introduce a contravariant idempotent adjunction between (i) the category of ranked monads on $\mathsf{Set}$; and (ii) the category of internal categories and internal retrofunctors in the category of locales. The left adjoint takes a monad $T$-viewed as a notion of computation, following Moggi-to its localic behaviour category $\mathsf{LB}T$. This behaviour category is understood as "the universal transition system" for interacting with $T$: its "objects" are states and the "morphisms" are transitions. On the other hand, the right adjoint takes a localic category $\mathsf{LC}$-similarly understood as a transition system-to the monad $Γ\mathsf{LC}$ where $(Γ\mathsf{LC})A$ is the set of $A$-indexed families of local sections to the source map which jointly partition the locale of objects. The fixed points of this adjunction consist of (i) hyperaffine-unary monads, i.e., those monads where term $t$ admits a read-only operation $\bar{t}$ predicting the output of $t$; and (ii) ample localic categories, i.e., whose source maps are local homeomorphisms and whose locale of objects are strongly zero-dimensional. The hyperaffine-unary monads arise in earlier works by Johnstone and Garner as a syntactic characterization of those monads with Cartesian closed Eilenberg-Moore categories. This equivalence is the Stone duality for monads; so-called because it further restricts to the classical Stone duality by viewing a Boolean algebra $B$ as a monad of $B$-partitions and the corresponding Stone space as a localic category with only identity morphisms.