Shapely monads and analytic functors

In mathematics and computer science, we often encounter structures which are faithfully encoded by a graphical calculus of the following sort. The basic data of the structure are depicted as certain diagrams; the basic operations of the structure act by glueing together these diagrams along certain parts of their boundaries; and the axioms of the structure are just those necessary to ensure that “every two ways of glueing a compound diagram together agree”. Commonly, such calculi depict structures wherein “functions”, “arrows” or “processes” are wired together along input or output “ports”. For instance, we have multicategories [21], whose arrows have many inputs but only one output; polycategories [29], whose arrows have multiple inputs and outputs, with composition subject to a linear wiring discipline; and coloured properads [31] and props [24], which are like polycategories but allow for non-linear wirings. Mathematical structures such as these are important in algebraic topology and homological algebra—encoding, for example, operations arising on infinite loop spaces [26] or on Hochschild cochains [27]—but also in logic and computer science. For example, polycategories encode the underlying semantics of a linear sequent calculus [22], while props have recently been used as an algebraic foundation for notions of computational network such as signal flow graphs [3] and Bayesian networks [10]. Other kinds of graphical structures arising in computer science include proof nets [11, §2], interaction nets [20] and bigraphs [14]. There is an established approach to describing structures of the above kind using monads on presheaf categories. The presheaf category captures the essential topology of the underlying graphical calculus, while the monad encodes both the wiring operations of the structure and the axioms that they obey; the algebras for the monad are the instances of the structure. One aspect which this approach does not account for is that the axioms should be determined by the requirement that “every two ways of wiring a compound diagram together agree”. The first main contribution of this paper is to rectify this: we explain the observed form of the axioms as a property of the associated monad—which we term shapeliness—stating that “every two operations of the same shape coincide”.