Specware supports the systematic construction of formal specifications and their stepwise refinement into programs. The fundamental operations in Specware are that of composing specifications (via colimits), the corresponding refinement by composing refinements (via sheaves), and the generation of programs by composing code modules (via colimits). The concept of diagram refinement is introduced as a practical realization of composing refinements via sheaves. Sequential and parallel composition of refinements satisfy a distributive law which is a generalization of similar compatibility laws in the literature. Specware is based on a rich categorical framework with a small set of orthogonal concepts. We believe that this formal basis will enable the scaling to system-level software construction.

Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of “geometric” models for concurrency. We show that the properties we want to prove on concurrent systems are stable under some form of deformation, which is almost homotopy. In fact, as the “direction ” of time matters, we have to allow deformation only as long as we do not reverse the direction of time. This calls for a new homotopy theory: “directed ” or di-homotopy. We develop some of the geometric intuition behind this theory and give some hints about the algebraic objects one can associate with it (in particular homology groups). For some historic as well as for some deeper reasons, the theory is at a stage where there is a nice blend between cubical, ω-categorical and topological techniques.

this paper are algebraic dcpos, and many of the points discussed here will be needed later in the special case. 2 They provide a simple example to illustrate the "Display categories" in Section 3.2

In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2-groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kan-fibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...

Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how this homotopy theory can be extended to the topos itself,!Smp. As a crucial advantage, the fundamental groupoid Π1:!Smp = Gpd is left adjoint to a natural functor M1: Gpd =!Smp, the symmetric nerve of a groupoid, and preserves all colimits – a strong van Kampen property. Similar results hold in all higher dimensions. Analogously, a notion of (non-reversible) directed homotopy can be developed in the ordinary simplicial topos Smp, with applications to image analysis as in [G3]. We have now a homotopy n-category functor ↑Πn: Smp = n-Cat, left adjoint to a nerve Nn = n-Cat(↑Πn(∆[n]), –). This construction can be applied to various presheaf categories; the basic requirements seem to be: finite products of representables are finitely presentable and there is a representable 'standard interval'.

Abstract We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of interest.

Abstract. Let G be a closed subgroup of Gn, the extended Morava stabilizer group. Let En be the Lubin-Tate spectrum, X an arbitrary spectrum with trivial G-action, and let ˆ L = L K(n). We prove that ˆ L(En ∧ X) is a continuous G-spectrum with homotopy fixed point spectrum ( ˆ L(En ∧ X)) hG, defined with respect to the continuous action. Also, we construct a descent spectral sequence whose abutment is π∗( ( ˆ L(En∧X)) hG). We show that the homotopy fixed points of ˆ L(En ∧ X) come from the K(n)-localization of the homotopy fixed points of the spectrum (Fn ∧ X). 1.

We present a truth-functional semantics for necessity-valued logics, based on the forcing technique. We interpret possibility distributions (which correspond to necessity measures) as informational states, and introduce a suitable language (basically, an extension of classical logic, similar to Pavelka’s language). Then we define the relation of “forcing ” between an informational state and a formula, meaning that the state contains enough information to support the validity of the formula. The subsequent step is the definition of a many-valued truth-functional semantics, by simply taking the truth value of a formula to be the set of all informational states that force the truth of the formula. A proof system in sequent calculus form is provided, and validity and completeness theorems are proved. 1

Restriction to geometric logic can enable one to define topological structures and continuous maps without explicit reference to topologies. This idea is illustrated with some examples and used to explain 1

We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. We test the notion in categories of interest. In the category of sets, any closed and bounded interval of real numbers is an interval object. In the category of topological spaces, the interval objects are closed and bounded intervals with the Euclidean topology. We also prove that an interval object exists in any elementary topos with natural numbers object. The universal property of an interval object provides a mechanism for defining functions on the interval. We use this to define basic arithmetic operations, and to verify equations between them. It also allows us to develop an analogue of the primitive recursive functions, yielding a natural class of computable functions on the interval. Contents 1