Metagories (Q2310774)

The present paper introduces what can be considered yet another categorical structure whose motivation comes from metric structures. This agrees with Lawvere's standpoint of viewing fundamental structures as categorical ones. The introduced categorical structure, called metagory (see Definition 3.3), is a generalization of the Aliouche-Simpson approximate categorical structures. The major point of the present generalization is to allow the ``area values'' to live in an arbitrary commutative an unital quantale \(V\) (and, in particular, allowing the area value \( +\infty \) in the case of the quantale \(V=\left[ 0, +\infty\right] \) which encompasses the Aliouche-Simpson approximate categorical structures). Within the more general context of \(V\)-metagories, the authors were able to greatly generalize and shed light over the previously introduced theories, neatly studying several categorical properties of the \(V\)-metagories. In particular, they gave a Yoneda-like embedding of a ``transitive \(V\)-metagory'' into a \(V\)-metric category, which encompasses what seems to be the main result of [\textit{A. Aliouche} and \textit{C. Simpson}, Theory Appl. Categ. 32, 1522--1562 (2017; Zbl 1408.18001)].