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Quasi-abelian categories and sheaves. (English) Zbl 0926.18004
Following D. Quillen’s approach to homological algebra on non-abelian categories, G. Laumon obtained interesting results for filtered \(\mathcal D\)-modules. One can also treat locally convex topological vector spaces in the same light. However, as the author observes, filtered \(\mathcal D\)-modules and locally convex topological vector spaces are examples of what the author terms quasi-abelian categories, which he develops as one develops the theory of abelian categories. Thus, the derived category of a quasi-abelian category and the attendant notions are introduced. For an elementary quasi-abelian category \(\mathcal E\), the category of sheaves with values in \(\mathcal E\) is almost as easy to regulate as a category of sheaves with values in an abelian category.

MSC:
18G50 Nonabelian homological algebra (category-theoretic aspects)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18E30 Derived categories, triangulated categories (MSC2010)
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