22 research outputs found
Towards a directed homotopy type theory
In this paper, we present a directed homotopy type theory for reasoning
synthetically about (higher) categories, directed homotopy theory, and its
applications to concurrency. We specify a new `homomorphism' type former for
Martin-L\"of type theory which is roughly analogous to the identity type former
originally introduced by Martin-L\"of. The homomorphism type former is meant to
capture the notions of morphism (from the theory of categories) and directed
path (from directed homotopy theory) just as the identity type former is known
to capture the notions of isomorphism (from the theory of groupoids) and path
(from homotopy theory). Our main result is an interpretation of these
homomorphism types into Cat, the category of small categories. There, the
interpretation of each homomorphism type hom(a,b) is indeed the set of
morphisms between the objects a and b of a category C. We end the paper with an
analysis of the interpretation in Cat with which we argue that our homomorphism
types are indeed the directed version of Martin-L\"of's identity types
From Cubes to Twisted Cubes via Graph Morphisms in Type Theory
Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure of higher groupoids. Bezem, Coquand, and Huber [Bezem et al., 2014] have presented a constructive model of univalence using a specific cube category, which we call the BCH cube category.
The higher categories encoded with the BCH cube category have the property that all morphisms are invertible, mirroring the fact that equality is symmetric. This might not always be desirable: the field of directed type theory considers a notion of equality that is not necessarily invertible.
This motivates us to suggest a category of twisted cubes which avoids built-in invertibility. Our strategy is to first develop several alternative (but equivalent) presentations of the BCH cube category using morphisms between suitably defined graphs. Starting from there, a minor modification allows us to define our category of twisted cubes. We prove several first results about this category, and our work suggests that twisted cubes combine properties of cubes with properties of globes and simplices (tetrahedra)
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A type theory for cartesian closed bicategories
We construct an internal language for cartesian closed bicategories. Precisely, we introduce a type theory modelling the structure of a cartesian closed bicategory and show that its syntactic model satisfies an appropriate universal property, thereby lifting the Curry-Howard-Lambek correspondence to the bicategorical setting. Our approach is principled and practical. Weak substitution structure is constructed using a bicategorification of the notion of abstract clone from universal algebra, and the rules for products and exponentials are synthesised from semantic considerations. The result is a type theory that employs a novel combination of 2-dimensional type theory and explicit substitution, and directly generalises the Simply-Typed Lambda Calculus. This work is the first step in a programme aimed at proving coherence for cartesian closed bicategories
A Cubical Implementation of Homotopical Patch Theory
We consider theoretical models of version control systems based on Homotopy Type Theory (HoTT). The main contribution is an implementation of Angiuli et al.âs Homotopical Patch Theory in Cubical Agda. Additionally the first chapter contains an approachable introduction to HoTT and Cubical Agda aimed at an audience of interested computer science students covering dependent Martin-Löf-style type theory, propositions as types, univalent foundations, higher inductive types and CCHM cubical type theory. Finally, we discuss some other approaches to a theory of version control systems in Darcsâ âalgebra of patchesâ and an unsuccessful attempt to model repositories in type theory as coequalizers.Master's Thesis in InformaticsINF399MAMN-INFMAMN-PRO
Transpension: The Right Adjoint to the Pi-type
Presheaf models of dependent type theory have been successfully applied to
model HoTT, parametricity, and directed, guarded and nominal type theory. There
has been considerable interest in internalizing aspects of these presheaf
models, either to make the resulting language more expressive, or in order to
carry out further reasoning internally, allowing greater abstraction and
sometimes automated verification. While the constructions of presheaf models
largely follow a common pattern, approaches towards internalization do not.
Throughout the literature, various internal presheaf operators (,
, , , ,
, the strictness axiom and locally fresh names) can be found and
little is known about their relative expressivenes. Moreover, some of these
require that variables whose type is a shape (representable presheaf, e.g. an
interval) be used affinely.
We propose a novel type former, the transpension type, which is right adjoint
to universal quantification over a shape. Its structure resembles a dependent
version of the suspension type in HoTT. We give general typing rules and a
presheaf semantics in terms of base category functors dubbed multipliers.
Structural rules for shape variables and certain aspects of the transpension
type depend on characteristics of the multiplier. We demonstrate how the
transpension type and the strictness axiom can be combined to implement all and
improve some of the aforementioned internalization operators (without formal
claim in the case of locally fresh names)
Decalf: A Directed, Effectful Cost-Aware Logical Framework
We present , a irected, ffectful ost-ware ogical ramework for studying
quantitative aspects of functional programs with effects. Like ,
the language is based on a formal phase distinction between the extension and
the intension of a program, its pure behavior as distinct from its cost
measured by an effectful step-counting primitive. The type theory ensures that
the behavior is unaffected by the cost accounting. Unlike , the
present language takes account of effects, such as probabilistic choice and
mutable state; this extension requires a reformulation of 's
approach to cost accounting: rather than rely on a "separable" notion of cost,
here a cost bound is simply another program. To make this formal, we equip
every type with an intrinsic preorder, relaxing the precise cost accounting
intrinsic to a program to a looser but nevertheless informative estimate. For
example, the cost bound of a probabilistic program is itself a probabilistic
program that specifies the distribution of costs. This approach serves as a
streamlined alternative to the standard method of isolating a recurrence that
bounds the cost in a manner that readily extends to higher-order, effectful
programs.
The development proceeds by first introducing the type system,
which is based on an intrinsic ordering among terms that restricts in the
extensional phase to extensional equality, but in the intensional phase
reflects an approximation of the cost of a program of interest. This
formulation is then applied to a number of illustrative examples, including
pure and effectful sorting algorithms, simple probabilistic programs, and
higher-order functions. Finally, we justify via a model in the
topos of augmented simplicial sets