Toposes pour les nuls

A brief introduction to Grothendieck's idea of toposes as generalized topological spaces, written from a topological viewpoint

On the specification of a component repository

The lack of a commonly accepted definition of a software component, the proliferation of competing `standards' and component frameworks, is here to stay, raising the fundamental question in component based development of how to cope in practice with heterogeneity. This paper reports on the design of a Component Repository aimed to give at least a partial answer to the above question. The repository was fully specified in Vdm and a working prototype is currently being used in an industrial environment

A hierarchical category model for geometrical product specifications (GPS)

International standards for tolerancing (ISO GPS) have undergone considerable evolutionary changes to meet the demands of the modern information age. Their expanding quantity and complexity have proposed a great obstacle to their informatisation progress. In this paper, a solution to reduce the complexity is coarse-graining the GPS knowledge into five hierarchy levels. A high-level abstraction mathematical theory - category theory is employed to model the GPS hierarchy, in which structures are modelled by categorical concepts such as categories, morphisms, pullbacks, functors and adjoint functors. As category theory is hierarchically structured itself, it can prove that the multi-level GPS framework is constructed in a rigorous manner and is expected to facilitate the future autonomous integration between design and measurement in the manufacturing system

Categorial L\'evy Processes

We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. The obtained independence for morphisms is called categorial independence. We define categorial L\'evy processes on every tensor category with initial unit object and present a construction generalizing the reconstruction of a L\'evy process from its convolution semigroup via the Daniell-Kolmogorov theorem. Finally, we discuss examples showing that many known independences from algebra as well as from (noncommutative) probability are special cases of categorial independence.Comment: Changes in v2: Abstract and introduction extended. Background on tensor functors moved to Section 2. Example section extended and reorganized. References updated. Acknowledgements updated. (Some Enrivonment numbers have changed!