More on soundness in the enriched context

TL;DR

This paper advances the theory of soundness in enriched category theory, exploring locally presentable categories, accessibility, and algebraic theories within this enriched framework.

Contribution

It extends soundness concepts to enriched categories, analyzing accessibility and algebraic theories for enriched monads, providing new theoretical insights.

Findings

Development of locally $ ext{Φ}$-presentable $ ext{V}$-categories

Conditions under which $ ext{Φ}$-accessible categories are $ ext{Ψ}$-accessible

A notion of $ ext{Φ}$-ary equational theory characterizing $ ext{Φ}$-monads

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 $\mathcal V$.