Comonads as Spaces

Monads on the category $\mathrm{𝐒𝐞𝐭}$ provide a simple and unifying notion of algebra: groups, rings, and vector spaces are all examples of algebraic structures corresponding to monads. In the upcoming work [3] we explore the dual idea that comonads provide a simple and unifying notion of space.

First, topological spaces are identified with certain comonads on $\mathrm{𝐒𝐞𝐭}$. We characterize these as precisely the density comonads of diagrams of subsets of a set (determining topological spaces as subbases). Thus the concept of topological space falls out of the theory of comonads, without mention of unions or finite intersections.

More general than topological spaces — but still not as general as arbitrary comonads on $\mathrm{𝐒𝐞𝐭}$ — are toposes equipped with a set of enough points [6, C2.2], or equivalently ionads as introduced by Garner [7]. Ionads at once generalize both topological spaces and small categories. An ionad amounts to a set $X$ and a finite-limit-preserving comonad on ${\mathrm{𝐒𝐞𝐭}}^X$. They may also be identified with pullback-preserving comonads on $\mathrm{𝐒𝐞𝐭}$ itself. In particular, this recovers Ahman and Uustalu’s result [2] that small categories are identified with certain comonads on $\mathrm{𝐒𝐞𝐭}$, namely those with polynomial carrier.

Following Garner’s insightful work, we generalize aspects of the theory of topological spaces to arbitrary comonads, not only on $\mathrm{𝐒𝐞𝐭}$, but on arbitrary categories. We give categorical definitions of basis, subbasis, and continuous map. We also define a double category of comonads on a fixed category, in which the two types of arrows are such continuous maps and ordinary comonad morphisms. Restricting to those comonads on $\mathrm{𝐒𝐞𝐭}$ corresponding to categories recovers the double category of functors and retrofunctors of Clarke and Di Meglio [4].

In the field of coalgebra [9], coalgebras are used to model transition systems, dynamical systems, automata, and various infinite data types [5]. Whereas monads correspond to varieties of algebras specified by equations as in universal algebra, comonads correspond to covarieties of coalgebras specified by coequations [1]. A potential application of the theory we develop is to approach universal coalgebra using tools from topology. As a start in this direction, we are able to recover a foundational assumption in coalgebra, namely the preservation of weak pullbacks [8], from a topological perspective. More precisely, we characterize weak-pullback-preserving comonads on $\mathrm{𝐒𝐞𝐭}$ as the density comonads of diagrams with co-confluent category of elements, a generalization of the ordinary definition of basis from topology, and we show that all such diagrams are bases according to our more abstract definition.