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Polynomial functors and polynomial monads
, 2009
"... Abstract. We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship wi ..."
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Abstract. We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship with operads and other related notions is explored.
2Dimensional Directed Type Theory
"... Recent work on higherdimensional type theory has explored connections between MartinLöf type theory, higherdimensional category theory, and homotopy theory. These connections suggest a generalization of dependent type theory to account for computationally relevant proofs of propositional equality ..."
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Recent work on higherdimensional type theory has explored connections between MartinLöf type theory, higherdimensional category theory, and homotopy theory. These connections suggest a generalization of dependent type theory to account for computationally relevant proofs of propositional equality—for example, taking IdSet A B to be the isomorphisms between A and B. The crucial observation is that all of the familiar type and term constructors can be equipped with a functorial action that describes how they preserve such proofs. The key benefit of higherdimensional type theory is that programmers and mathematicians may work up to isomorphism and higher equivalence, such as equivalence of categories. In this paper, we consider a further generalization of higherdimensional type theory, which associates each type with a directed notion of transformation between its elements. Directed type theory accounts for phenomena not expressible in symmetric higherdimensional type theory, such as a universe set of sets and functions, and a type Ctx used in functorial abstract syntax. Our formulation requires two main ingredients: First, the types themselves must be reinterpreted to take account of variance; for example, a Π type is contravariant in its domain, but covariant in its range. Second, whereas in symmetric type theory proofs of equivalence can be internalized using the MartinLöf identity type, in directed type theory the twodimensional structure must be made explicit at the judgemental level. We describe a 2dimensional directed type theory, or 2DTT, which is validated by an interpretation into the strict 2category Cat of categories, functors, and natural transformations. We also discuss applications of 2DTT for programming with abstract syntax, generalizing the functorial approach to syntax to the dependently typed and mixedvariance case. 1
A Categorical Treatment of Ornaments
"... Abstract—Ornaments aim at taming the multiplication of specialpurpose datatypes in dependently typed programming languages. In type theory, purpose is logic. By presenting datatypes as the combination of a structure and a logic, ornaments relate these specialpurpose datatypes through their common ..."
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Abstract—Ornaments aim at taming the multiplication of specialpurpose datatypes in dependently typed programming languages. In type theory, purpose is logic. By presenting datatypes as the combination of a structure and a logic, ornaments relate these specialpurpose datatypes through their common structure. In the original presentation, the concept of ornament was introduced concretely for an example universe of inductive families in type theory, but it was clear that the notion was more general. This paper digs out the abstract notion of ornaments in the form of a categorical model. As a necessary first step, we abstract the universe of datatypes using the theory of polynomial functors. We are then able to characterise ornaments as cartesian morphisms between polynomial functors. We thus gain access to powerful mathematical tools that shall help us understand and develop ornaments. We shall also illustrate the adequacy of our model. Firstly, we rephrase the standard ornamental constructions into our framework. Thanks to its conciseness, we gain a deeper understanding of the structures at play. Secondly, we develop new ornamental constructions, by translating categorical structures into type theoretic artefacts.
CATEGORICAL LOGIC AND PROOF THEORY EPSRC INDIVIDUAL GRANT REPORT – GR/R95975/01
"... Abstract. I describe the main results obtained during the EPSRC postdoctoral fellowship that I held at the University of Cambridge. The fellowship focused on the interplay between category theory and mathematical logic. 1. Wellfounded trees Wtypes in categories. Types of wellfounded trees, or Wtyp ..."
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Abstract. I describe the main results obtained during the EPSRC postdoctoral fellowship that I held at the University of Cambridge. The fellowship focused on the interplay between category theory and mathematical logic. 1. Wellfounded trees Wtypes in categories. Types of wellfounded trees, or Wtypes, are one of the most important components of MartinLöf’s dependent type theories. They allow us to define a wide class of inductive types, play an essential role in the ‘setsastrees’ interpretation of constructive set theories, and contribute considerably to the prooftheoretic strength of dependent type theories. A categorical counterpart of Wtypes was introduced in [18] by defining Wtypes in a locally cartesian closed category to be initial algebras for endofunctors of a special kind, generally referred to as polynomial functors. In collaboration with Martin Hyland, I set out to investigate the consequences of the assumption that a locally cartesian closed category has Wtypes. To explore these consequences we introduced the notion of a dependent polynomial functor, a
Constructive Membership and Indexes in Trees
, 2009
"... Trees carrying information stored in their nodes are a fundamental abstract data type. Approaching trees in a formal constructive environment allows us to realize properties of trees, inherent in their structure. Specifically we will look at the evidence provided by the predicates which operate on t ..."
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Trees carrying information stored in their nodes are a fundamental abstract data type. Approaching trees in a formal constructive environment allows us to realize properties of trees, inherent in their structure. Specifically we will look at the evidence provided by the predicates which operate on these trees. This evidence, expressed in terms of logical and programming languages, is realizable only in a constructive context. In the constructive setting, membership predicates over recursive types are inhabited by terms indexing the elements that satisfy the criteria for membership. In this paper, we motivate and explore this idea in the concrete setting of lists and trees. We first provide a background in constructive type theory and show relavent properties of trees. We present and define the concept of inhabitants of a generic shape type that corresponds naturally and exactly to the inhabitants of a membership predicate. In this context, (λx.T rue) ∈ S is the set of all indexes into S, but we show that not all subsets of indexes are expressible by strictly local predicates. Accordingly, we extend our membership predicates to predicates that compute and hold the state “from above” as well as allow “looking below”. The modified predicates of this form are complete in the sense that they can express every subset of indexes in S. These ideas are motivated by experience programming in Nuprl’s constructive type theory and the theorems for lists and trees have been formalized and mechanically checked. 1
BOOTSTRAPPING INDUCTIVE AND COINDUCTIVE TYPES IN HASCASL
, 812
"... Abstract. We discuss the treatment of initial datatypes and final process types in the widespectrum 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 constru ..."
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Abstract. We discuss the treatment of initial datatypes and final process types in the widespectrum 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.
BOOTSTRAPPING INDUCTIVE AND COINDUCTIVE TYPES IN HASCASL
, 2007
"... Vol. 4 (4:17) 2008, pp. 1–27 ..."
Small Induction Recursion
"... Abstract. There are several different approaches to the theory of data types. At the simplest level, polynomials and containers give a theory of data types as free standing entities. At a second level of complexity, dependent polynomials and indexed containers handle more sophisticated data types in ..."
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Abstract. There are several different approaches to the theory of data types. At the simplest level, polynomials and containers give a theory of data types as free standing entities. At a second level of complexity, dependent polynomials and indexed containers handle more sophisticated data types in which the data have an associated indices which can be used to store important computational information. The crucial and salient feature of dependent polynomials and indexed containers is that the index types are defined in advance of the data. At the most sophisticated level, inductionrecursion allows us to define data and indices simultaneously. This work investigates the relationship between the theory of small inductive recursive definitions and the theory of dependent polynomials and indexed containers. Our central result is that the expressiveness of small inductive recursive definitions is exactly the same as that of dependent polynomials and indexed containers. A second contribution of this paper is the definition of morphisms of small inductive recursive definitions. This allows us to extend our main result to an equivalence
Small Induction Recursion, Indexed Containers and Dependent Polynomials are equivalent ∗
"... There are several different approaches to the theory of data types. At the simplest level, polynomials and containers give a theory of data types as free standing entities. At a second level of complexity, dependent polynomials and indexed containers handle more sophisticated data types in which the ..."
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There are several different approaches to the theory of data types. At the simplest level, polynomials and containers give a theory of data types as free standing entities. At a second level of complexity, dependent polynomials and indexed containers handle more sophisticated data types in which the data have an associated indices which can be used to store important computational information. The crucial and salient feature of dependent polynomials and indexed containers is that the index types are defined in advance of the data. At the most sophisticated level, inductionrecursion allows us to define the data and the indices simultaneously. The aim of this work is to investigate the relationship between the theory of small inductive recursive definitions and the theory of dependent polynomials and indexed containers. Our central result is that the expressiveness of small inductive recursive definitions is exactly the same as that of dependent polynomials and indexed containers. Formally, this result applies not just to the data types definable in these theories, but also to the morphisms between such data