Large limit sketches and topological space objects

TL;DR

This paper explores large limit sketches and their relation to topological space objects, providing new universal characterizations of categories like Top and their functors, extending classical results in topology and category theory.

Contribution

It introduces a framework connecting limit sketches with topological and cotopological space objects, generalizing Lawvere theories and offering new proofs of Isbell's classification.

Findings

Equivalence between cocontinuous functors and models of the opposite sketch.

Universal properties of cocomplete categories modeled by large limit sketches.

Characterization of Top as both a cocomplete and complete category.

Abstract

For a (possibly large) realized limit sketch $\mathcal{S}$ such that every $\mathcal{S}$-model is small in a suitable sense we show that the category of cocontinuous functors $\mathsf{Mod}(\mathcal{S}) \to \mathcal{C}$ into a cocomplete category $\mathcal{C}$ is equivalent to the category $\mathsf{Mod}_{\mathcal{C}}(\mathcal{S}^{\mathrm{op}})$ of $\mathcal{C}$-valued $\mathcal{S}^{\mathrm{op}}$-models. From this result we deduce universal properties of several examples of cocomplete categories appearing in practice. It can be applied in particular to infinitary Lawvere theories, generalizing the well-known case of finitary Lawvere theories. We also look at a large limit sketch that models $\mathsf{Top}$, study the corresponding notion of an internal net-based topological space object, and deduce from our main result that cocontinuous functors $\mathsf{Top} \to \mathcal{C}$ into a cocomplete category $\mathcal{C}$ correspond to net-based cotopological space objects internal to $\mathcal{C}$. Finally, we describe a limit sketch that models $\mathsf{Top}^{\mathrm{op}}$ and deduce from our main result that continuous functors $\mathsf{Top} \to \mathcal{C}$ into a complete category $\mathcal{C}$ correspond to frame-based topological space objects internal to $\mathcal{C}$. Thus, we characterize $\mathsf{Top}$ both as a cocomplete and as a complete category. Thereby we get two new conceptual proofs of Isbell's classification of cocontinuous functors $\mathsf{Top} \to \mathsf{Top}$ in terms of topological topologies.