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Derived categories and universal problems. (English) Zbl 0722.18002

The origin of this paper is the question whether the canonical embedding \({\mathcal A}\to {\mathcal D}^ b{\mathcal A}\) of an exact category (in the sense of Quillen) into its bounded derived category is ‘universal’ among the functors from \({\mathcal A}\) to triangulated categories which transform exact sequences to triangles. Unfortunately, this is not the case. However, we are able to prove a universal property for the system of functors \({\mathcal A}^{\wedge}\to {\mathcal D}^ b{\mathcal A}^{\wedge}\), where \({\mathcal A}^{\wedge}\) is the sequence \((=tower)\) of the categories formed by the commutative n-cubes in \({\mathcal A}\) and \({\mathcal D}^ b{\mathcal A}^{\wedge}\) the tower of their derived categories. The main ingredient of the proof is a description of \({\mathcal D}^ b{\mathcal A}\) as a localization of a category of presheaves. This description also yields a characterization of the ‘construction \({\mathcal D}^ b\)’ among all 2-functors from the 2-category of exact categories to the 2-category of triangulated categories.
Reviewer: B.Keller

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
18E35 Localization of categories, calculus of fractions
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
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References:

[1] Beilinson A. A., in Ktheory, arithmetic and geometry 1289 (1987)
[2] Beilinson A. A., Faisceauz pervers 100 (1982)
[3] DOI: 10.2307/2372628 · Zbl 0050.17205 · doi:10.2307/2372628
[4] Gabriel P., Representation theory · Zbl 0482.16026
[5] Gabriel P., Calculw of Fractions and Homotopy theory 1 (1967) · doi:10.1007/978-3-642-85844-4
[6] Gray J. W., Formal category theory: Adjointness for 2-categories 391 (1974) · Zbl 0285.18006 · doi:10.1007/BFb0061280
[7] DOI: 10.1007/BF02564452 · Zbl 0626.16008 · doi:10.1007/BF02564452
[8] DOI: 10.2307/1970153 · Zbl 0084.26704 · doi:10.2307/1970153
[9] Keller B., Chain complezes and stable categories
[10] Keller B., C. R. Acad. Sci.Paris, Série I 305 pp 225– (1987)
[11] MacLane S., Graduate Texts in Mathematics 5 (1971)
[12] Quillen, D. 1973. ”Higher Algebraic K-theory I”. Vol. 341, 85–147. Springer LNM. · Zbl 0292.18004
[13] DOI: 10.1112/jlms/s2-39.3.436 · Zbl 0642.16034 · doi:10.1112/jlms/s2-39.3.436
[14] DOI: 10.1016/0022-4049(89)90081-9 · Zbl 0685.16016 · doi:10.1016/0022-4049(89)90081-9
[15] Verdier J.-L., état 0, SGA 4 1/2 569 pp 262– (1977)
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