Extensivity of categories of relational structures

TL;DR

This paper demonstrates that categories of models for certain relational Horn theories are infinitely extensive, including categories like preorders, posets, and various metric spaces, with explicit characterizations of embeddings and quotients.

Contribution

It establishes the infinite extensivity of categories of models for relational Horn theories under mild conditions, extending known examples and providing explicit characterizations of categorical constructs.

Findings

Categories of models of certain relational Horn theories are infinitely extensive.

Explicit descriptions of initial sources and final sinks, including embeddings and quotients.

Includes categories like preorders, posets, and various metric space categories.

Abstract

We prove that the category of models of any relational Horn theory satisfying a mild syntactic condition is infinitely extensive. Central examples of such categories include the categories of preordered sets and partially ordered sets, and the categories of small $\mathscr{V}$-categories, (symmetric) pseudo-$\mathscr{V}$-metric spaces, and (symmetric) $\mathscr{V}$-metric spaces for a commutative unital quantale $\mathscr{V}$. We also explicitly characterize initial sources and final sinks in such categories, and in particular embeddings and quotients.