Diagrammatic Presentations of Enriched Monads and Varieties for a Subcategory of Arities

The theory of presentations of enriched monads was developed by Kelly, Power, and Lack, following classic work of Lawvere, and has been generalized to apply to subcategories of arities in recent work of Bourke–Garner and the authors. We argue that, while theoretically elegant and structurally fundamental, such presentations of enriched monads can be inconvenient to construct directly in practice, as they do not directly match the definitional procedures used in constructing many categories of enriched algebraic structures via operations and equations. Retaining the above approach to presentations as a key technical underpinning, we establish a flexible formalism for directly describing enriched algebraic structure borne by an object of a \(\mathscr {V}\)-category \(\mathscr {C}\) in terms of parametrized \(\mathscr {J}\)-ary operations and diagrammatic equations for a suitable subcategory of arities \(\mathscr {J}\hookrightarrow \mathscr {C}\). On this basis we introduce the notions of diagrammatic \(\mathscr {J}\)-presentation and \(\mathscr {J}\)-ary variety, and we show that the category of \(\mathscr {J}\)-ary varieties is dually equivalent to the category of \(\mathscr {J}\)-ary \(\mathscr {V}\)-monads. We establish several examples of diagrammatic \(\mathscr {J}\)-presentations and \(\mathscr {J}\)-ary varieties relevant in both mathematics and theoretical computer science, and we define the sum and tensor product of diagrammatic \(\mathscr {J}\)-presentations. We show that both \(\mathscr {J}\)-relative monads and \(\mathscr {J}\)-pretheories give rise to diagrammatic \(\mathscr {J}\)-presentations that directly describe their algebras. Using diagrammatic \(\mathscr {J}\)-presentations as a method of proof, we generalize the pretheories-monads adjunction of Bourke and Garner beyond the locally presentable setting. Lastly, we generalize Birkhoff’s Galois connection between classes of algebras and sets of equations to the above setting.