Van Kampen theorems for categories of covering morphisms in lextensive categories
The authors use a formulation of the Van Kampen theorem in terms of fundamental groupoids and covering spaces to describe an appropriate categorical setting for this type of result. The key ingredients in their formulation are the notions of ``lextensive'' category (i.e., a category with finite limits and finite coproducts that are disjoint and universal) and of effective descent morphism. The ``open covering'' being of effective descent turns out to be equivalent to the generalized Van Kampen theorem. This is a considerable improvement over the classical situation, where only sufficient homotopical conditions for the theorem were known. This very elegant result can then be used not just for topological coverings, but also in algebraic settings. A number of examples are supplied.
scientific article; zbMATH DE number 1038604
scientific article