More on soundness in the enriched context

Abstract

Working within enriched category theory, we further develop the use of soundness, introduced by Ad\'amek, Borceux, Lack, and Rosick\'y for ordinary categories. In particular we investigate: (1) the theory of locally $\Phi$-presentable $\mathcal V$-categories for a sound class $\Phi$, (2) the problem of whether every $\Phi$-accessible $\mathcal V$-category is $\Psi$-accessible, for given sound classes $\Phi\subseteq\Psi$, and (3) a notion of $\Phi$-ary equational theory whose $\mathcal V$-categories of models characterize algebras for $\Phi$-ary monads on $\mathcal V$.