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Exact categories. (English) Zbl 1192.18007
This very useful treatise develops ab ovo the theory of exact categories in the sense of Quillen. All fundamental properties are deduced in a detailed, yet elegant manner from the axioms of an exact category. Nonetheless, in App. A a proof of the embedding theorem of Gabriel-Quillen-Laumon is given, which characterises exact categories as extension-closed subcategories of (possibly big) abelian categories.
Exact categories arise in contexts similar to or derived from abelian ones, but where either there are not enough short exact sequences, or where a consideration of all short exact sequences would yield undesirable consequences, so that it is advisable to fix a set of short exact sequences with good properties; its elements are called admissible. A typical example is the category of complexes with values in an abelian category. One declares only pointwise split short exact sequences to be admissible. As a consequence, split acyclic complexes become injective and projective relative to this set. So the resulting exact category is even Frobenius; its stable category is the usual homotopy category of complexes.
In the introductory §1, a wealth of examples is mentioned. A historical note explains the genesis of exact categories; in particular, we learn that Yoneda knew exact categories already in 1960. More examples and historical comments can be found throughout the text, in particular in §13.
In §2, the cokernel criterion for certain commutative quadrangles to be bicartesian (with admissible diagonal) is given.
In §3, the $$3\times 3$$-lemma is proven.
In §4, it is shown that quasi-abelian categories (every morphism has a kernel, kernels are stable under pushout; and dually) are exact.
In §5, it is sketched how to use functors to produce new exact categories from given ones.
In §6, it is sketched that the idempotent completion of an exact category is exact.
In §7, weakly idempotent complete exact categories (every retraction has a kernel; or dually) are shown to be the suitable class of exact categories to be considered for certain elementary properties.
In §8, the circumference lemma (kernel-cokernel-sequence) and the snake lemma are proven in weakly idempotent complete exact categories.
In §9 and §10, the derived category of an exact category and derived functors in the sense of Grothendieck-Deligne-Keller are discussed.
In §11 and §12, the connection to derived functors in the classical sense is explained, where we count exact categories as classical.
§13 is devoted to examples and applications, e.g. Banach spaces.
In App. A, a proof of the Gabriel-Quillen-Laumon embedding theorem is given, following the lines of Laumon and Thomason, using sheaves on sites. The assertion is that the Yoneda functor embeds an exact category $$\mathcal{A}$$ as an extension-closed full subcategory into the abelian category of left-exact functors from $$\mathcal{A}^\circ$$ to abelian groups. Moreover, a sequence in $$\mathcal{A}$$ is admissibly short exact if and only if its image is short exact. Finally, if $$\mathcal{A}$$ is weakly idempotent complete, then this embedding reflects admissible epics. This is useful in practice if one wants to use abelian category lemmas for exact category assertions. The disadvantage is that the abelian category used to embed in is somewhat difficult to control.
Finally, whereas the author does not claim originality, quite a few arguments appear to be original, and the presentation as a whole surely is. This treatise should have appeared as a textbook.

##### MSC:
 18E10 Abelian categories, Grothendieck categories 18E30 Derived categories, triangulated categories (MSC2010) 18-02 Research exposition (monographs, survey articles) pertaining to category theory
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