14 research outputs found
Dependent Inductive and Coinductive Types are Fibrational Dialgebras
In this paper, I establish the categorical structure necessary to interpret
dependent inductive and coinductive types. It is well-known that dependent type
theories \`a la Martin-L\"of can be interpreted using fibrations. Modern
theorem provers, however, are based on more sophisticated type systems that
allow the definition of powerful inductive dependent types (known as inductive
families) and, somewhat limited, coinductive dependent types. I define a class
of functors on fibrations and show how data type definitions correspond to
initial and final dialgebras for these functors. This description is also a
proposal of how coinductive types should be treated in type theories, as they
appear here simply as dual of inductive types. Finally, I show how dependent
data types correspond to algebras and coalgebras, and give the correspondence
to dependent polynomial functors.Comment: In Proceedings FICS 2015, arXiv:1509.0282
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
W-types in Homotopy Type Theory
We will give a detailed account of why the simplicial sets model of the
univalence axiom due to Voevodsky also models W-types. In addition, we will
discuss W-types in categories of simplicial presheaves and an application to
models of set theory.Comment: We have corrected the statement of Theorem 3.4. We thank Christian
Sattler for alerting us to the error in the original versio
Models of Non-Well-Founded Sets via an Indexed Final Coalgebra Theorem
The paper uses the formalism of indexed categories to recover the proof of a
standard final coalgebra theorem, thus showing existence of final coalgebras
for a special class of functors on categories with finite limits and colimits.
As an instance of this result, we build the final coalgebra for the powerclass
functor, in the context of a Heyting pretopos with a class of small maps. This
is then proved to provide a model for various non-well-founded set theories,
depending on the chosen axiomatisation for the class of small maps
Non-Wellfounded Trees in Homotopy Type Theory
Coinductive data types are used in functional programming to represent infinite data struc-tures. Examples include the ubiquitous data type of streams over a given base type, but also more sophisticated types. From a categorical perspective, coinductive types are characterized by a universal property, which specifies the object with that property uniquely in a suitable sense. More precisely, a coinductive type is specified as the terminal coalgebra of a suitable endofunctor. In this category-theoretic viewpoint, coinductive types are dual to inductive types, which are defined as initial algebras. Inductive, resp. coinductive, types are usually considered in the principled form of the family of W-types, resp. M-types, parametrized by a type A and a dependent type family B over A, that is, a family of types (B(a))a:A. Intuitively, the elements of the coinductive type M(A,B) are trees with nodes labeled by elements of A such that a node labeled by a: A has B(a)-many subtrees, given by a map B(a) → M(A,B); see Figure 1 for an example. The inductive type W(A,B) contains only trees where any path within that tree eventually leads to a leaf, that is, to a node a: A such that B(a) is empty. a, b, c: A B(a) =
Bootstrapping Inductive and Coinductive Types in HasCASL
We discuss the treatment of initial datatypes and final process types in the
wide-spectrum language HasCASL. In particular, we present specifications that
illustrate how datatypes and process types arise as bootstrapped concepts using
HasCASL's type class mechanism, and we describe constructions of types of
finite and infinite trees that establish the conservativity of datatype and
process type declarations adhering to certain reasonable formats. The latter
amounts to modifying known constructions from HOL to avoid unique choice; in
categorical terminology, this means that we establish that quasitoposes with an
internal natural numbers object support initial algebras and final coalgebras
for a range of polynomial functors, thereby partially generalising
corresponding results from topos theory. Moreover, we present similar
constructions in categories of internal complete partial orders in
quasitoposes