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- W4300580713 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." @default.
- W4300580713 created "2022-10-03" @default.
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- W4300580713 date "2015-12-18" @default.
- W4300580713 modified "2023-09-27" @default.
- W4300580713 title "Shapely monads and analytic functors" @default.
- W4300580713 doi "https://doi.org/10.48550/arxiv.1512.05980" @default.
- W4300580713 hasPublicationYear "2015" @default.
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