France arrondissements, which are, in turn, subdivided into 2,054 cantons. These cantons are then divided into 36,658 communes, which are municipalities with an elected municipal council. Three communes—Paris, Lyon and Marseille—are subdivided into 45 municipal arrondissements. The regions, departments and communes are all known as territorial collectivities, meaning they possess local assemblies as well as an executive. Arrondissements and cantons are merely administrative divisions. However, this was not always the case. Until 1940, the arrondissements were territorial collectivities with an elected assembly, but these were suspended by the Vichy regime and definitely abolished by the Fourth Republic in 1946. In addition
Subdivided interval categories Subdivided interval categories In category theory (mathematics) there exists an important collection of categories denoted formula_1 for natural numbers formula_2. The objects of formula_3 are the integers formula_4, and the morphism set formula_5 for objects formula_6 is empty if formula_7. Subdivided interval categories are very useful in defining simplicial sets. The category whose objects are the subdivided interval categories and whose morphisms are functors is often written formula_8 and is called the simplicial indexing category. A simplicial set is just a contravariant functor formula_9. The category 𝟘 is an empty interval, that is, an empty category, having no objects nor