In this article, 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 â € 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 generalization 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 characterized 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 article, we do this for some of the examples listed above; in future work, we intend to use this to obtain canonical notions of denotational model for graphical calculi such as Milner's bigraphs, Lafont's interaction nets or Girard's multiplicative proof nets.
- analytic functors
- graphical calculi