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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 ‘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.

Shapely monads and analytic functors