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Lawvere's algebraic theories, or Lawvere theories, underpin a categoricalapproach to general algebra, and Lawvere's adjunction between semantics andalgebraic structure leads to an equivalence between Lawvere theories andfinitary monads on the category of sets. Several authors have transported theseideas to a variety of settings, including contexts of category theory enrichedin a symmetric monoidal closed category. In this paper, we develop a generalaxiomatic framework for enriched structure-semantics adjunctions andmonad-theory equivalences for subcategories of arities. Not only do weestablish a simultaneous generalization of the monad-theory equivalencespreviously developed in the settings of Lawvere (1963), Linton (1966), Dubuc(1970), Borceux-Day (1980), Power (1999), Nishizawa-Power (2009),Lack-Rosick\'y (2011), Lucyshyn-Wright (2016), and Bourke-Garner (2019), butalso we establish a structure-semantics theorem that generalizes those given inthe first four of these works while applying also to the remaining five, forwhich such a result has not previously been developed. Furthermore, we employour axiomatic framework to establish broad new classes of examples of enrichedmonad-theory equivalences and structure-semantics adjunctions for subcategoriesof arities enriched in locally bounded closed categories, including variousconvenient closed categories that are relevant in topology and analysis andneed not be locally presentable.

Enriched structure-semantics adjunctions and monad-theory equivalences for subcategories of arities