806 research outputs found
Data types with symmetries and polynomial functors over groupoids
Polynomial functors are useful in the theory of data types, where they are
often called containers. They are also useful in algebra, combinatorics,
topology, and higher category theory, and in this broader perspective the
polynomial aspect is often prominent and justifies the terminology. For
example, Tambara's theorem states that the category of finite polynomial
functors is the Lawvere theory for commutative semirings. In this talk I will
explain how an upgrade of the theory from sets to groupoids is useful to deal
with data types with symmetries, and provides a common generalisation of and a
clean unifying framework for quotient containers (cf. Abbott et al.), species
and analytic functors (Joyal 1985), as well as the stuff types of Baez-Dolan.
The multi-variate setting also includes relations and spans, multispans, and
stuff operators. An attractive feature of this theory is that with the correct
homotopical approach - homotopy slices, homotopy pullbacks, homotopy colimits,
etc. - the groupoid case looks exactly like the set case. After some standard
examples, I will illustrate the notion of data-types-with-symmetries with
examples from quantum field theory, where the symmetries of complicated tree
structures of graphs play a crucial role, and can be handled elegantly using
polynomial functors over groupoids. (These examples, although beyond species,
are purely combinatorial and can be appreciated without background in quantum
field theory.) Locally cartesian closed 2-categories provide semantics for
2-truncated intensional type theory. For a fullfledged type theory, locally
cartesian closed \infty-categories seem to be needed. The theory of these is
being developed by D.Gepner and the author as a setting for homotopical
species, and several of the results exposed in this talk are just truncations
of \infty-results obtained in joint work with Gepner. Details will appear
elsewhere.Comment: This is the final version of my conference paper presented at the
28th Conference on the Mathematical Foundations of Programming Semantics
(Bath, June 2012); to appear in the Electronic Notes in Theoretical Computer
Science. 16p
Applicative Bidirectional Programming with Lenses
A bidirectional transformation is a pair of mappings between source and view data objects, one in each direction. When the view is modified, the source is updated accordingly with respect to some laws. One way to reduce the development and maintenance effort of bidirectional transformations is to have specialized languages in which the resulting programs are bidirectional by construction---giving rise to the paradigm of bidirectional programming.
In this paper, we develop a framework for applicative-style and higher-order bidirectional programming, in which we can write bidirectional transformations as unidirectional programs in standard functional languages, opening up access to the bundle of language features previously only available to conventional unidirectional languages. Our framework essentially bridges two very different approaches of bidirectional programming, namely the lens framework and Voigtlander’s semantic bidirectionalization, creating a new programming style that is able to bag benefits from both
Higher Order Containers
Abstract. Containers are a semantic way to talk about strictly positive types. In previous work it was shown that containers are closed under various constructions including products, coproducts, initial algebras and terminal coalgebras. In the present paper we show that, surprisingly, the category of containers is cartesian closed, giving rise to a full cartesian closed subcategory of endofunctors. The result has interesting applications syntax. We also show that while the category of containers has finite limits, it is not locally cartesian closed.
Towards a Convenient Category of Topological Domains
We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models
A Convenient Category of Domains
We motivate and define a category of "topological domains",
whose objects are certain topological spaces, generalising
the usual -continuous dcppos of domain theory.
Our category supports all the standard constructions of domain theory,
including the solution of recursive domain equations. It also
supports the construction of free algebras for (in)equational
theories, provides a model of parametric polymorphism,
and can be used as the basis for a theory of computability.
This answers a question of Gordon Plotkin, who asked
whether it was possible to construct a category of domains
combining such properties
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