Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations

This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category \Tops of stratified spaces, that are topological spaces $X$ endowed with a partition \cF and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element (X,\cF) of \Tops together with a class \cA of subsets of $X$; they are similar to invariants introduced by M. Clapp and D. Puppe. If (X,\cF)\in\Tops, we define a transverse subset as a subspace $A$ of $X$ such that the intersection $S\cap A$ is at most countable for any S\in \cF. Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a $C^1$-foliation, the three previous definitions, with \cA the class of transverse subsets, coincide with the tangential category and are homotopical invariants.Comment: 14 pages, 2 figure

Scala.js: Type-Directed Interoperability with Dynamically Typed Languages

Interoperability between statically typed and dynamically typed languages is increasingly important, as can be witnessed by the many statically typed languages targeting JavaScript. Interoperating with both the object-oriented and functional features of JavaScript is essential, if only to manipulate the DOM, yet existing languages have very poor support for this. We present Scala.js, a dialect of Scala compiling to JavaScript. Its interoperability system is based on a powerful and intuitive framework for type-directed interoperability with dynamically typed languages. The framework combines facade types for JavaScript values; user-defined, implicit, type-directed cross-language conversions; and a Dynamic type building on facade types and implicit conversions. It accommodates both the functional and object-oriented features of Scala and JavaScript, and provides very natural interoperability between the two languages. It is expressive enough to represent the DOM and jQuery APIs, among others, both in its statically typed and dynamically typed flavors

Cross-Platform Language Design

Programming languages are increasingly compiled to multiple runtimes, each featuring their own rich structures such as their object model. Furthermore, they need to interact with other languages targeting said runtimes. A language targeting only one runtime can be designed to tailor its semantics to those of that runtime, for easy interoperability with other languages. However, in a language targeting multiple runtimes with differing semantics, it is difficult to cater to each of them while retaining a common behavior across runtimes. We call \emph{cross-platform language} a language that aims at being both \emph{portable} across platforms and \emph{interoperable} with each target platform. Portability is the ability for a program or a library to cross-compile for multiple platforms, and behave the same way on all of them. Interoperability is the ability to communicate with other languages on the same platform. While many cross-compiling languages focus on one of these two properties---only adding support for the other one as an afterthought---, languages that are designed from the ground up to support both are rare. In this thesis, we present the design of Scala.js, the dialect of Scala targeting the JavaScript platform, which turned Scala into a cross-platform language. On the one hand, Scala programs can be cross-compiled for the JVM and JavaScript with portable semantics. On the other hand, whereas Scala/JVM interoperates with Java, Scala.js interoperates with JavaScript, allowing to use any JavaScript library. Along the dissertation, we give insights that can be transferred to the design of other cross-platform languages, although with a bias towards those targeting the JVM and JavaScript. The first and most obvious challenge is to reconcile the static nature of Scala's object model with JavaScript's dynamic one. Besides the ability to mutate a class hierarchy at run-time in JavaScript, there are fundamental differences between the two models, in particular the difference between compile-time overloading and run-time overloading. We discuss how such semantic mismatches can live in harmony within the language. The second challenge is to obtain good performance from a language where interoperability with a dynamic and unknown part of the program is pervasive. To that end, we design and specify an intermediate representation (IR) with first-class support for dynamically typed interoperability features in addition to statically typed JVM-style operations. Despite its tight integration with the open world of JavaScript, most of the IR can be considered as a closed world where advanced whole-program optimizations can be performed. The performance of the overall system is evaluated and shown to be competitive with hand-written JavaScript, and even with Scala/JVM in some cases

Inductive Lusternik-Schnirelmann category in a model category

We introduce the notion of inductive category in a model category and prove that it agrees with the Ganea approach given by Doeraene. This notion also coincides with the topological one when we consider the category of (well-) pointed topological spaces.Comment: 14 page