Tameness in generalized metric structures

TL;DR

This paper extends the framework of metric abstract elementary classes by allowing metrics to take values in a quantale, demonstrating that tameness results hold in this broader setting and exploring further generalizations.

Contribution

It generalizes metric AECs to quantale-valued metrics and confirms tameness results in this new context, also considering partial metrics and potential connections to fuzzy structures.

Findings

Tameness results extend to quantale-valued metric AECs.

Framework accommodates partial metric spaces.

Suggests links to fuzzy structures and sheaves.

Abstract

We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the metric to take values in a well-behaved quantale. As a proof of concept we show that the result of Boney and Zambrano on (metric) tameness under a large cardinal assumption holds in this more general context. We briefly consider a further generalization to partial metric spaces, and hint at connections to classes of fuzzy structures, and structures on sheaves.