Wellfounded Trees and Dependent Polynomial Functors

Gambino, Nicola; Hyland, Martin

doi:10.1007/978-3-540-24849-1_14

Cited by 78 publications

(107 citation statements)

References 14 publications

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“…[5], [6].) The forgetful functor PolyMnd(I) → PolyEnd(I) has a left adjoint, denoted P → P. The monad P is the free monad on P.…”

Section: Propositionmentioning

confidence: 99%

“…Trees cannot be captured by such, since it is essential to be able to distinguish the edges in a tree. The case of arbitrary polynomial functors was considered by Gambino and Hyland [5], corresponding to dependent type theory.…”

Section: Theoremmentioning

confidence: 99%

“…The theory of polynomial functors is relatively new and has hitherto mostly been explored from the viewpoint of type theory and computer science (some references can be found in [5] or [12]). The point that polynomial functors are an excellent tool for making explicit and analysing the combinatorics underlying operad theory was first made in the paper [13], where polynomial functors were used to extract the first purely combinatorial characterisation of opetopes.…”

Section: Introductionmentioning

confidence: 99%

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Polynomial Functors and Trees

Kock

2010

International Mathematics Research Notices

152

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We explore the relationship between polynomial functors and (rooted) trees. In the first part we use polynomial functors to derive a new convenient formalism for trees, and obtain a natural and conceptual construction of the category Ω of Moerdijk and Weiss; its main properties are described in terms of some factorisation systems. Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets. In the second part we describe polynomial endofunctors and monads as structures built from trees, characterising the images of several nerve functors from polynomial endofunctors and monads into presheaves on categories of trees. Polynomial endofunctors and monads over a base are characterised by a sheaf condition on categories of decorated trees. In the absolute case, one further condition is needed, a certain projectivity condition, which serves also to characterise polynomial endofunctors and monads among (coloured) collections and operads.

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“…[5], [6].) The forgetful functor PolyMnd(I) → PolyEnd(I) has a left adjoint, denoted P → P. The monad P is the free monad on P.…”

Section: Propositionmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Polynomial Functors and Trees

Kock

2010

International Mathematics Research Notices

152

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“…It is di cult to trace back its origin since this concept permeates the history of proof theory and a large part of theoretical computer science In recent years, the desire to explore, understand, and extend the concept of an inductive definition has led di↵erent researchers to di↵erent but (extensionally) equivalent notions. The theory of containers [1], and polynomial functors [18,12] are some of the outcomes of this research These theories give a comprehensive account of those data types such as Nat (the natural numbers), List a (lists containing data of a given type a), and Tree a (trees containing, once more, data of a given type a) which are free-standing in that their definition does not require the definition of other inter-related data types.…”

Section: Introductionmentioning

confidence: 99%

“…vectors which record the lengths of lists. Therefore containers and polynomials have been generalised to indexed containers [2] and dependent polynomials [12,13] to capture not only free standing data types such as those mentioned above, but also data types where the data are indexed by an index storing computationally relevant information. Containers and (non-dependent) polynomials arise as specific instances of these generalised notions where the type of indices is chosen to be a singleton type.…”

Section: Introductionmentioning

confidence: 99%

Small Induction Recursion

Hancock

McBride

Ghani

et al. 2013

Lecture Notes in Computer Science

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Abstract. There are several di↵erent 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, induction-recursion 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 between the category of small inductive recursive definitions and the category of dependent polynomials/indexed containers. We comment on both the theoretical and practical ramifications of this result.

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Container Combinatorics: Monads and Lax Monoidal Functors

Uustalu

2017

Topics in Theoretical Computer Science

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Abbott et al.'s containers are a "syntax" for a wide class of set functors in terms of shapes and positions. Containers whose "denotation" carries a comonad structure can be characterized as directed containers, or containers where a shape and a position in it determine another shape, intuitively a subshape of this shape rooted by this position. In this paper, we develop similar explicit characterizations for container functors with a monad structure and container functors with a lax monoidal functor structure as well as some variations. We argue that this type of characterizations make a tool, e.g., for enumerating the monad structures or lax monoidal functors that some set functor admits. Such explorations are of interest, e.g., in the semantics of effectful functional programming languages.

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Wellfounded Trees and Dependent Polynomial Functors

Cited by 78 publications

References 14 publications

Polynomial Functors and Trees

Polynomial Functors and Trees

Small Induction Recursion

Container Combinatorics: Monads and Lax Monoidal Functors

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