Injective Spaces via Adjunction

TL;DR

This paper explores the categorical structure of topological spaces, showing that cocomplete spaces are exactly the injective spaces and form algebras for a specific monad, using enriched category theory concepts.

Contribution

It establishes a categorical framework connecting cocompleteness, injectivity, and monads in the context of topological theories, extending known results.

Findings

Cocomplete spaces are precisely the injective spaces.

Injective spaces are algebras for a specific monad on Set.

Enriched category theory provides new insights into topological spaces.

Abstract

Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence relation $\mathfrak{x}\to x$ between ultrafilters and points of a topological space $X$ as arrows in $X$. Naturally, this point of view opens the door to the use of concepts and ideas from (enriched) Category Theory for the investigation of (for instance) topological spaces. In this paper we study cocompleteness, adjoint functors and Kan extensions in the context of topological theories. We show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on $\SET$. This way we obtain enriched versions of known results about injective topological spaces and continuous lattices.