107 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
Fibrational induction rules for initial algebras
This paper provides an induction rule that can be used to prove properties of data structures whose types are inductive, i.e., are carriers of initial algebras of functors. Our results are semantic in nature and are inspired by Hermida and Jacobsâ elegant algebraic formulation of induction for polynomial data types. Our contribution is to derive, under slightly different assumptions, an induction rule that is generic over all inductive types, polynomial or not. Our induction rule is generic over the kinds of properties to be proved as well: like Hermida and Jacobs, we work in a general fibrational setting and so can accommodate very general notions of properties on inductive types rather than just those of particular syntactic forms. We establish the correctness of our generic induction rule by reducing induction to iteration. We show how our rule can be instantiated to give induction rules for the data types of rose trees, finite hereditary sets, and hyperfunctions. The former lies outside the scope of Hermida and Jacobsâ work because it is not polynomial; as far as we are aware, no induction rules have been known to exist for the latter two in a general fibrational framework. Our instantiation for hyperfunctions underscores the value of working in the general fibrational setting since this data type cannot be interpreted as a set
Generic Fibrational Induction
This paper provides an induction rule that can be used to prove properties of
data structures whose types are inductive, i.e., are carriers of initial
algebras of functors. Our results are semantic in nature and are inspired by
Hermida and Jacobs' elegant algebraic formulation of induction for polynomial
data types. Our contribution is to derive, under slightly different
assumptions, a sound induction rule that is generic over all inductive types,
polynomial or not. Our induction rule is generic over the kinds of properties
to be proved as well: like Hermida and Jacobs, we work in a general fibrational
setting and so can accommodate very general notions of properties on inductive
types rather than just those of a particular syntactic form. We establish the
soundness of our generic induction rule by reducing induction to iteration. We
then show how our generic induction rule can be instantiated to give induction
rules for the data types of rose trees, finite hereditary sets, and
hyperfunctions. The first of these lies outside the scope of Hermida and
Jacobs' work because it is not polynomial, and as far as we are aware, no
induction rules have been known to exist for the second and third in a general
fibrational framework. Our instantiation for hyperfunctions underscores the
value of working in the general fibrational setting since this data type cannot
be interpreted as a set.Comment: For Special Issue from CSL 201
Non-wellfounded trees in Homotopy Type Theory
We prove a conjecture about the constructibility of coinductive types - in
the principled form of indexed M-types - in Homotopy Type Theory. The
conjecture says that in the presence of inductive types, coinductive types are
derivable. Indeed, in this work, we construct coinductive types in a subsystem
of Homotopy Type Theory; this subsystem is given by Intensional Martin-L\"of
type theory with natural numbers and Voevodsky's Univalence Axiom. Our results
are mechanized in the computer proof assistant Agda.Comment: 14 pages, to be published in proceedings of TLCA 2015; ancillary
files contain Agda files with formalized proof
Indexed Induction and Coinduction, Fibrationally
This paper extends the fibrational approach to induction and coinduction
pioneered by Hermida and Jacobs, and developed by the current authors, in two
key directions. First, we present a dual to the sound induction rule for
inductive types that we developed previously. That is, we present a sound
coinduction rule for any data type arising as the carrier of the final
coalgebra of a functor, thus relaxing Hermida and Jacobs' restriction to
polynomial functors. To achieve this we introduce the notion of a quotient
category with equality (QCE) that i) abstracts the standard notion of a
fibration of relations constructed from a given fibration; and ii) plays a role
in the theory of coinduction dual to that played by a comprehension category
with unit (CCU) in the theory of induction. Secondly, we show that inductive
and coinductive indexed types also admit sound induction and coinduction rules.
Indexed data types often arise as carriers of initial algebras and final
coalgebras of functors on slice categories, so we give sufficient conditions
under which we can construct, from a CCU (QCE) U:E \rightarrow B, a fibration
with base B/I that models indexing by I and is also a CCU (resp., QCE). We
finish the paper by considering the more general case of sound induction and
coinduction rules for indexed data types when the indexing is itself given by a
fibration
Indexed induction and coinduction, fibrationally.
This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a sound coinduction rule for any data type arising as the final coalgebra of a functor, thus relaxing Hermida and Jacobsâ restriction to polynomial data types. For this we introduce the notion of a quotient category with equality (QCE), which both abstracts the standard notion of a fibration of relations constructed from a given fibration, and plays a role in the theory of coinduction dual to that of a comprehension category with unit (CCU) in the theory of induction. Second, we show that indexed inductive and coinductive types also admit sound induction and coinduction rules. Indexed data types often arise as initial algebras and final coalgebras of functors on slice categories, so our key technical results give sufficent conditions under which we can construct, from a CCU (QCE) U : E -> B, a fibration with base B/I that models indexing by I and is also a CCU (QCE)
Infinite Types, Infinite Data, Infinite Interaction
We describe a way to represent computable functions between coinductive types as particular transducers in type theory. This generalizes earlier work on functions between streams by P. Hancock to a much richer class of coinductive types. Those transducers can be defined in dependent type theory without any notion of equality but require inductive-recursive definitions. Most of the properties of these constructions only rely on a mild notion of equality (intensional equality) and can thus be formalized in the dependently typed language Agda
Representing Continuous Functions between Greatest Fixed Points of Indexed Containers
We describe a way to represent computable functions between coinductive types
as particular transducers in type theory. This generalizes earlier work on
functions between streams by P. Hancock to a much richer class of coinductive
types. Those transducers can be defined in dependent type theory without any
notion of equality but require inductive-recursive definitions. Most of the
properties of these constructions only rely on a mild notion of equality
(intensional equality) and can thus be formalized in the dependently typed
language Agda
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
A Linear Category of Polynomial Diagrams
We present a categorical model for intuitionistic linear logic where objects
are polynomial diagrams and morphisms are simulation diagrams. The
multiplicative structure (tensor product and its adjoint) can be defined in any
locally cartesian closed category, whereas the additive (product and coproduct)
and exponential Tensor-comonoid comonad) structures require additional
properties and are only developed in the category Set, where the objects and
morphisms have natural interpretations in terms of games, simulation and
strategies.Comment: 20 page
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