Shapely monads and analytic functors

TL;DR

This paper introduces shapely monads as a mathematical framework for structures represented by graphical calculi, characterizing their operations and axioms through analytic functors and providing a basis for canonical denotational models.

Contribution

It formalizes shapely monads within the context of analytic functors, connecting graphical calculi with algebraic structures and offering a method to generate structures directly from their graphical syntax.

Findings

Shapely monads are characterized as submonads of a universal analytic monad.

Shapely monads correspond to structures with exactly one operation per shape.

The framework enables defining data and axioms directly from graphical calculi.

Abstract

In this paper, we give precise mathematical form to the idea of a structure whose data and axioms are faithfully represented by a graphical calculus; some prominent examples are operads, polycategories, properads, and PROPs. Building on the established presentation of such structures as algebras for monads on presheaf categories, we describe a characteristic property of the associated monads---the shapeliness of the title---which says that "any two operations of the same shape agree". An important part of this work is the study of analytic functors between presheaf categories, which are a common generalisation of Joyal's analytic endofunctors on sets and of the parametric right adjoint functors on presheaf categories introduced by Diers and studied by Carboni--Johnstone, Leinster and Weber. Our shapely monads will be found among the analytic endofunctors, and may be characterised as the submonads of a universal analytic monad with "exactly one operation of each shape". In fact, shapeliness also gives a way to define the data and axioms of a structure directly from its graphical calculus, by generating a free shapely monad on the basic operations of the calculus. In this paper we do this for some of the examples listed above; in future work, we intend to do so for graphical calculi such as Milner's bigraphs, Lafont's interaction nets, or Girard's multiplicative proof nets, thereby obtaining canonical notions of denotational model.