739 research outputs found
Monads need not be endofunctors
We introduce a generalization of monads, called relative monads, allowing for underlying functors between different categories. Examples include finite-dimensional vector spaces, untyped and typed λ-calculus syntax and indexed containers. We show that the Kleisli and Eilenberg-Moore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions, relative monads are monoids in the functor category concerned and extend to monads, giving rise to a coreflection between relative monads and monads. Arrows are also an instance of relative monads
Shapely monads and analytic functors
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
Monads in Double Categories
We extend the basic concepts of Street's formal theory of monads from the
setting of 2-categories to that of double categories. In particular, we
introduce the double category Mnd(C) of monads in a double category C and
define what it means for a double category to admit the construction of free
monads. Our main theorem shows that, under some mild conditions, a double
category that is a framed bicategory admits the construction of free monads if
its horizontal 2-category does. We apply this result to obtain double
adjunctions which extend the adjunction between graphs and categories and the
adjunction between polynomial endofunctors and polynomial monads.Comment: 30 pages; v2: accepted for publication in the Journal of Pure and
Applied Algebra; added hypothesis in Theorem 3.7 that source and target
functors preserve equalizers; on page 18, bottom, in the statement concerning
the existence of a left adjoint, "if and only if" was replaced by "a
sufficient condition"; acknowledgements expande
Lie monads and dualities
We study dualities between Lie algebras and Lie coalgebras, and their
respective (co)representations. To allow a study of dualities in an
infinite-dimensional setting, we introduce the notions of Lie monads and Lie
comonads, as special cases of YB-Lie algebras and YB-Lie coalgebras in additive
monoidal categories. We show that (strong) dualities between Lie algebras and
Lie coalgebras are closely related to (iso)morphisms between associated Lie
monads and Lie comonads. In the case of a duality between two Hopf algebras -in
the sense of Takeuchi- we recover a duality between a Lie algebra and a Lie
coalgebra -in the sense defined in this note- by computing the primitive and
the indecomposables elements, respectively.Comment: 27 pages, v2: some examples added and minor change
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