In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences ...

Working in the context of symmetric spectra, we consider any higher algebraic structures that can be described as algebras over an operad O. We prove that the fundamental adjunction comparing O-algebra spectra with coalgebra spectra over the associated comonad K, via topological Quillen homology (or TQ-homology), can be turned into an equivalence of homotopy theories by replacing O-algebras wit...

The Curry-Howard correspondence between formulas and types, proofs and programs, proof simplification and program execution, also holds for intuitionistic modal logic S4. It turns out that the S4 modalities translate as a monoidal comonad on the space of proofs, giving rise to a canonical augmented simplicial structure. We study the geometry of these augmented simplicial sets, showing that each...

process categories (APCs) are axiomatically defined categorical models of FRP with processes, which we developed recently [7]. They are an extension of temporal categories, which are models of FRP with behaviors and events, but without processes [5]. In this section, we give an introduction to APCs. 3.1 The Basics An APC is a category C with some additional structure. FRP types are modeled by o...

This thesis studies problems of higher-order model-checking from a semantic and logical perspective. Higher-order model-checking is concerned with the verification of properties expressed in monadic second-order logic, specified over infinite trees generated by a class of rewriting systems called higher-order recursion schemes. These systems are equivalent to simply-typed λ-terms with recursion...

Containers [1] are an elegant representation of a wide class of datatypes in terms of shapes and positions in shapes. In our FoSSaCS 2012 work [2], we introduced directed containers as a special case to account for the common situation where every position in a shape determines another shape, informally the subshape rooted by that position; some examples being the datatypes of nonempty lists an...

Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known...

This paper presents a work whose aim is to obtain information on execution time of λ-terms by semantic means. By execution time, we mean the number of steps in a computational model. As in [Ehrhard and Regnier 2006], the computational model considered in this paper will be Krivine’s machine, a more realistic model than β-reduction. Indeed, Krivine’s machine implements (weak) head linear reducti...

This paper presents a work whose aim is to obtain information on execution time of λ-terms by semantic means. By execution time, we mean the number of steps in a computational model. As in [Ehrhard and Regnier 2006], the computational model considered in this paper will be Krivine’s machine, a more realistic model than β-reduction. Indeed, Krivine’s machine implements (weak) head linear reducti...

We present results concerning the solution of recursive domain equations in the category G of games, which is a modiied version of the category presented in AJM94]. New constructions corresponding to lifting and separated sum for games are presented, and are used to generate games for two simple recursive types: the vertical and lazy natural numbers. Recently, the \game semantics" paradigm has ...