More on soundness in the enriched context
Giacomo Tendas
TL;DR
This work extends the theory of soundness from ordinary to enriched categories, developing a robust framework for locally $\Phi$-presentable $\mathcal V$-categories under weakly sound weight classes. It establishes a suite of equivalent characterizations for local presentability, analyzes how to raise accessibility indices via a sharpness relation $\trianglelefteq$, and develops a universal-algebraic account via $\Phi$-ary languages and equational theories whose models correspond to algebras for $\Phi$-ary monads preserving $\Phi$-flat colimits. A key technical achievement is the enriched Gabriel–Ulmer duality and the demonstration that, for suitable $\Phi$, one can obtain $\Phi$-presentable, $\mathcal V$-projective generators and represent model categories as $\operatorname{Mod}(\mathbb E)$ or $\operatorname{Alg}(T)$. The results specialize to familiar settings (e.g., finite products, sifted colimits) and apply to enriched categories such as topological or categorical structures, offering a unified approach to enriched local presentability, accessibility, and universal algebra with potential for broad applications in enriched category theory and its foundations.
Abstract
Working within enriched category theory, we further develop the use of soundness, introduced by Adámek, Borceux, Lack, and Rosický for ordinary categories. In particular we investigate: (1) the theory of locally $Φ$-presentable $\mathcal V$-categories for a sound class $Φ$, (2) the problem of whether every $Φ$-accessible $\mathcal V$-category is $Ψ$-accessible, for given sound classes $Φ\subseteqΨ$, and (3) a notion of $Φ$-ary equational theory whose $\mathcal V$-categories of models characterize algebras for $Φ$-ary monads on $\mathcal V$.