Presentations and algebraic colimits of enriched monads for a subcategory of arities

Abstract

We develop a general framework for studying signatures, presentations, and algebraic colimits of enriched monads for a subcategory of arities, even when the base of enrichment $\mathcal{V}$ is not locally presentable. When $\mathcal{V}$ satisfies the weaker requirement of local boundedness, the resulting framework is sufficiently general to apply to the $\Phi$-accessible monads of Lack and Rosick\'y and the $\mathcal{J}$-ary monads of the first author, while even without local boundedness our framework captures in full generality the presentations of strongly finitary monads of Lack and Kelly as well as Wolff's presentations of $\mathcal{V}$-categories by generators and relations. Given any small subcategory of arities $j : \mathcal{J} \hookrightarrow \mathcal{C}$ in an enriched category $\mathcal{C}$, satisfying certain assumptions, we prove results on the existence of free $\mathcal{J}$-ary monads, the monadicity of $\mathcal{J}$-ary monads over $\mathcal{J}$-signatures, and the existence of algebraic colimits of $\mathcal{J}$-ary monads. We study a notion of presentation for $\mathcal{J}$-ary monads and show that every such presentation presents a $\mathcal{J}$-ary monad. Certain of our results generalize earlier results of Kelly, Power, and Lack for finitary enriched monads in the locally finitely presentable setting, as well as analogous results of Kelly and Lack for strongly finitary monads on cartesian closed categories. Our main results hold for a wide class of subcategories of arities in locally bounded enriched categories.