Metagories

TL;DR

Metagories are metrically enriched graphs with a structure assigning values to triangles, generalizing metric categories and enabling embeddings into enriched categories, with applications beyond classical metrics.

Contribution

This work generalizes recent theories by providing conditions for Yoneda-type embeddings of metagories into enriched categories, broadening applicability.

Findings

Established conditions for Yoneda-type embeddings.

Proved isometric embeddability into metrically enriched categories.

Extended the framework beyond classical metric spaces.

Abstract

Metric approximate categories, or metagories, for short, are metrically enriched graphs. Their structure assigns to every directed triangle in the graph a value which may be interpreted as the area of the triangle; alternatively, as the distance of a pair of consecutive arrows to any potential candidate for their composite. These values may live in an arbitrary commutative quantale. Generalizing and extending recent work by Aliouche and Simpson, we give a condition for the existence of an Yoneda-type embedding which, in particular, gives the isometric embeddability of a metagory into a metrically enriched category. The generality of the value quantale allows for applications beyond the classical metric context.