Injective objects and retracts of Fra\"iss\'e limits

TL;DR

This paper provides a category-theoretic characterization of retracts of Fra"iss"e limits, linking injectivity concepts to these structures and applying the results to Urysohn's universal metric space.

Contribution

It introduces a natural form of injectivity relative to category pairs to characterize retracts of Fra"iss"e limits, offering new insights into their structure.

Findings

Retracts of Fra"iss"e limits are exactly objects injective relative to certain category pairs.

The characterization applies to non-expansive retracts of Urysohn's universal metric space.

Provides a purely categorical perspective on classical structures in model theory and metric geometry.

Abstract

We present a purely category-theoretic characterization of retracts of Fra\"iss\'e limits. For this aim, we consider a natural version of injectivity with respect to a pair of categories (a category and its subcategory). It turns out that retracts of Fra\"iss\'e limits are precisely the objects that are injective relatively to such a pair. One of the applications is a characterization of non-expansive retracts of Urysohn's universal metric space.