Enriched Locally Generated Categories
Ivan Di Liberti, Jiří Rosický
TL;DR
The paper develops the notion of $\mathcal{M}$-locally generated categories for a factorization system $(\mathcal{E},\mathcal{M})$, and extends Gabriel-Ulmer duality to this setting via $\lambda$-nests and their models, including an enriched version over a closed base $\mathcal{V}$. It proves that suitable nests yield a duality between locally $\lambda$-generated categories and nests, and provides a concrete enriched application to Banach spaces by showing $\operatorname{Ban}$ is equivalent to the category of models of the nest $\operatorname{Ban}_{\text{fd}}^{op}$. The Horace-like framework clarifies how finite-dimensional Banach spaces generate the ambient Banach space category via multiple pullbacks of isometries and finite limits, within the enriched setting of $\mathbf{CMet}$. Overall, the work merges sketch-theoretic and enriched category theory to yield a versatile approach for analyzing both abstract categorical structure and concrete functional-analytic categories such as Banach spaces.
Abstract
We introduce the notion of $\mathcal{M}$-locally generated category for a factorization system $(\mathcal{E},\mathcal{M})$ and study its properties. We offer a Gabriel-Ulmer duality for these categories, introducing the notion of nest. We develop this theory also from an enriched point of view. We apply this technology to Banach spaces showing that it is equivalent to the category of models of the nest of finite-dimensional Banach spaces.