The enriched Vietoris monad on representable spaces

TL;DR

This paper introduces an 'up-set monad' on representable spaces, explores its properties, and connects categorical notions to classical topology, particularly relating to the Vietoris monad and complete spaces.

Contribution

It constructs and analyzes the 'up-set monad' on representable spaces, showing it is of Kock-Zöberlein type and linking categorical concepts to classical topological results.

Findings

The 'up-set monad' is of Kock-Zöberlein type.

The paper introduces a notion of weighted limit for enriched categories.

Connections are made between the monad and the upper Vietoris monad.

Abstract

Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual(?) notion of completeness does not behave as the mirror image of the one of cocompleteness; and in this paper we have a closer look at complete spaces. In particular, we construct the "up-set monad" on representable spaces (in the sense of L. Nachbin for topological spaces, respectively C. Hermida for multicategories); we show that this monad is of Kock-Z\"oberlein type; we introduce and study a notion of weighted limit similar to the classical notion for enriched categories; and we describe the Kleisli category of our "up-set monad". We emphasize that these generic categorical notions and results can be indeed connected to more "classical" topology: for topological spaces, the "up-set monad" becomes the upper Vietoris monad, and the statement "$X$ is totally cocomplete if and only if $X^\mathrm{op}$ is totally complete" specialises to O. Wyler's characterisation of the algebras of the Vietoris monad on compact Hausdorff spaces.