Shapely monads and analytic functors
Résumé
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 \emph{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 \emph{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 \emph{universal} analytic monad with “exactly one
operation of each shape”.
In fact, shapeliness also gives a way to \emph{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.
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