Shapely monads and analytic functors

• Published in: Journal of logic and computation 2018-02, Vol.28 (1), p.33-83
• Main Authors: Garner, Richard, Hirschowitz, Tom
• Format: Article
• Language: English
• Subjects: Algebraic Topology; Category Theory; Computer Science; Logic in Computer Science; Mathematics
• ISSN: 0955-792X; 1465-363X
• DOI: 10.1093/logcom/exx029

Abstract 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.