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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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##### References:
  M. Artin, Grothendieck Topologies, Mimeographed Notes, Harvard University, Cambridge, MA, 1962.  P. Balmer, Witt groups, in: Handbook of $$K$$-theory, vols. 1, 2, Springer, Berlin, 2005, pp. 539-576. MR2181829 (2006h:19004).  Barr, M., Exact categories, Lecture notes in mathematics, vol. 236, (1973), Springer Berlin  A.A. Beı˘linson, J. Bernstein, P. Deligne, Faisceaux pervers, Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque, vol. 100, Société Mathématique de France, Paris, 1982, pp. 5-171. MR751966 (86g:32015).  F. Borceux, Handbook of Categorical Algebra. 1, Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge, 1994. MR1291599 (96g:18001a).  F. Borceux, Handbook of Categorical Algebra. 2, Encyclopedia of Mathematics and its Applications, vol. 51, Cambridge University Press, Cambridge, 1994. MR1313497 (96g:18001b).  F. Borceux, Handbook of Categorical Algebra. 3, Encyclopedia of Mathematics and its Applications, vol. 52, Cambridge University Press, Cambridge, 1994. MR1315049 (96g:18001c).  Borel, A.; Grivel, P.-P.; Kaup, B.; Haefliger, A.; Malgrange, B.; Ehlers, F., Algebraic $$D$$-modules, Perspectives in mathematics, vol. 2, (1987), Academic Press Inc. Boston, MA, MR882000 (89g:32014)  A. Borel, N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. MR1721403 (2000j:22015). · Zbl 0980.22015  Buchsbaum, D.A., A note on homology in categories, Ann. math. (2), 69, 66-74, (1959), MR0140556 (25 #3974) · Zbl 0084.26801  T. Bühler, On the algebraic foundation of bounded cohomology, Ph.D. Thesis, ETH Zürich, 2008.  M. Bunge, Categories of set-valued functors, Ph.D. Thesis, University of Pennsylvania, 1966.  Butler, M.C.R.; Horrocks, G., Classes of extensions and resolutions, Philos. trans. roy. soc. London ser. A, 254, 155-222, (1961/1962), MR0188267 (32 #5706) · Zbl 0099.25902  Cartan, H.; Eilenberg, S., Homological algebra, (), MR1731415 (2000h:18022)  Dräxler, P.; Reiten, I.; Smalø, S.O.; Solberg, Ø., Exact categories and vector space categories, Trans. amer. math. soc., 351, 2, 647-682, (1999), (With an appendix by B. Keller. MR1608305 (99f:16001)) · Zbl 0916.16002  Freyd, P., Relative homological algebra made absolute, Proc. nat. acad. sci. USA, 49, 19-20, (1963), MR0146234 (26 #3756) · Zbl 0114.01403  Freyd, P., Abelian categories. an introduction to the theory of functors, (), MR0166240 (29 #3517)  Freyd, P., Representations in abelian categories, (), 95-120, MR0209333 (35 #231) · Zbl 0202.32402  Freyd, P., Splitting homotopy idempotents, (), 173-176, MR0206069 (34 #5894)  E.M. Friedlander, D.R. Grayson (Eds.), Handbook of $$K$$-theory. vols. 1, 2, Springer, Berlin, 2005. MR2182598 (2006e:19001).  Gabriel, P.; Roı˘ter, A.V., Representations of finite-dimensional algebras, Algebra, VIII, encyclopaedia mathematical sciences, vol. 73, (1992), Springer Berlin, (With a chapter by B. Keller, pp. 1-177. MR1239447 (94h:16001b)) · Zbl 0839.16001  Gabriel, P., Des catégories abéliennes, Bull. soc. math. France, 90, 323-448, (1962), MR0232821 (38 #1144) · Zbl 0201.35602  Gelfand, S.I.; Manin, Y.I., Methods of homological algebra, (), MR1950475 (2003m:18001) · Zbl 1006.18001  Grothendieck, Alexander, Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957) 119-221 MR0102537 (21 #1328).  Happel, D., Triangulated categories in the representation theory of finite-dimensional algebras, London mathematical society lecture note series, vol. 119, (1988), Cambridge University Press Cambridge, MR935124 (89e:16035) · Zbl 0635.16017  Helemskiı˘, A.Ya., The homology of Banach and topological algebras, Mathematics and its applications (soviet series), vol. 41, (1989), Kluwer Academic Publishers Group Dordrecht, MR1093462 (92d:46178)  Heller, A., Homological algebra in abelian categories, Ann. math. (2), 68, 484-525, (1958), MR0100622 (20 #7051) · Zbl 0084.26704  Heller, A., The loop-space functor in homological algebra, Trans. amer. math. soc., 96, 382-394, (1960), MR0116045 (22 #6840) · Zbl 0096.25502  Hilton, P.J.; Stammbach, U., A course in homological algebra, Graduate texts in mathematics, vol. 4, (1997), Springer New York, MR1438546 (97k:18001) · Zbl 0863.18001  Hochschild, G., Relative homological algebra, Trans. amer. math. soc., 82, 246-269, (1956), MR0080654 (18,278a) · Zbl 0070.26903  Hochschild, G.; Mostow, G.D., Cohomology of Lie groups, Illinois J. math., 6, 367-401, (1962), MR0147577 (26 #5092) · Zbl 0111.03302  Hoffmann, N.; Spitzweck, M., Homological algebra with locally compact abelian groups, Adv. math., 212, 2, 504-524, (2007), MR2329311 (2009d:22006) · Zbl 1123.22002  Johnson, B.E., Introduction to cohomology in Banach algebras, (), 84-100, MR0417787 (54 #5835)  Karoubi, M., Algèbres de Clifford et $$K$$-théorie, Ann. sci. école norm. sup. (4), 1, 161-270, (1968), MR0238927 (39 #287) · Zbl 0194.24101  Kashiwara, M.; Schapira, P., Categories and sheaves, Grundlehren der mathematischen wissenschaften [fundamental principles of mathematical sciences], vol. 332, (2006), Springer Berlin, MR2182076 (2006k:18001) · Zbl 1118.18001  Kechris, A.S., Classical descriptive set theory, Graduate texts in mathematics, vol. 156, (1995), Springer New York, MR1321597 (96e:03057) · Zbl 0819.04002  Keller, B., Chain complexes and stable categories, Manuscripta math., 67, 4, 379-417, (1990), MR1052551 (91h:18006) · Zbl 0753.18005  Keller, B., Derived categories and universal problems, Comm. algebra, 19, 3, 699-747, (1991), MR1102982 (92b:18010) · Zbl 0722.18002  B. Keller, Derived categories and their uses, in: Handbook of algebra, vol. 1, North-Holland, Amsterdam, 1996, pp. 671-701. MR1421815 (98h:18013). · Zbl 0862.18001  B. Keller, On differential graded categories, in: International Congress of Mathematicians, vol. II, European Mathematical Society, Zürich, 2006, pp. 151-190. MR2275593 (2008g:18015). · Zbl 1140.18008  Keller, B.; Vossieck, D., Sous LES catégories dérivées, C. R. acad. sci. Paris Sér. I math., 305, 6, 225-228, (1987), MR907948 (88m:18014) · Zbl 0628.18003  Künzer, M., Heller triangulated categories, Homology homotopy appl., 9, 2, 233-320, (2007), MR2366951 (2009b:18017) · Zbl 1128.18008  G. Laumon, Sur la catégorie dérivée des $$\mathcal{D}$$-modules filtrés, in: Algebraic Geometry (Tokyo/Kyoto, 1982), Lecture Notes in Mathematics, vol. 1016, Springer, Berlin, 1983, pp. 151-237. MR726427 (85d:32022).  Mac Lane, S., Homology, (), MR0156879 (28 #122)  Mac Lane, S., Categories for the working Mathematician, Graduate texts in mathematics, vol. 5, (1998), Springer New York, MR1712872 (2001j:18001) · Zbl 0906.18001  Mac Lane, S.; Moerdijk, I., Sheaves in geometry and logic, (), (A first introduction to topos theory), Corrected reprint of the 1992 edition. MR1300636 (96c:03119) · Zbl 0822.18001  Mitchell, B., The full imbedding theorem, Amer. J. math., 86, 619-637, (1964), MR0167511 (29 #4783) · Zbl 0124.01502  Neeman, A., The derived category of an exact category, J. algebra, 135, 2, 388-394, (1990), MR1080854 (91m:18016) · Zbl 0753.18004  Neeman, A., Triangulated categories, Annals of mathematics studies, vol. 148, (2001), Princeton University Press Princeton, NJ, MR1812507 (2001k:18010) · Zbl 0974.18008  Prosmans, F., Derived categories for functional analysis, Publ. res. inst. math. sci., 36, 1, 19-83, (2000), MR1749013 (2001g:46156) · Zbl 0973.46069  D. Quillen, Higher algebraic $$K$$-theory. I, in: Algebraic $$K$$-theory, I: Higher $$K$$-theories, Proceedings of the Conference, Battelle Memorial Institute, Seattle, Washington, 1972, Lecture Notes in Mathematics, vol. 341, Springer, Berlin, 1973, pp. 85-147. MR0338129 (49 #2895).  Rump, W., Almost abelian categories, Cahiers topologie Géom. différentielle catég., 42, 3, 163-225, (2001), MR1856638 (2002m:18008) · Zbl 1004.18009  Rump, W., A counterexample to Raikov’s conjecture, Bull. London math. soc., 40, 985-994, (2008) · Zbl 1210.18010  M. Schlichting, Hermitian $$K$$-theory of exact categories, J. $$K$$-theory, preprint (2008) 1-49, to appear.  J.-P. Schneiders, Quasi-abelian categories and sheaves, Mém. Soc. Math. Fr. (N.S.) 76 (1999) vi+134. MR1779315 (2001i:18023).  SGA4, Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Springer, Berlin, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, J.L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne, B. Saint-Donat, Lecture Notes in Mathematics, vol. 269, Springer, Berlin, MR0354652 (50 #7130).  Srinivas, V., Algebraic $$K$$-theory, Progress in mathematics, vol. 90, (1996), Birkhäuser Boston Inc. Boston, MA, MR1382659 (97c:19001) · Zbl 0860.19001  R.W. Thomason, T. Trobaugh, Higher algebraic $$K$$-theory of schemes and of derived categories, The Grothendieck Festschrift, vol. III, Progress in Mathematics, vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247-435. MR1106918 (92f:19001).  J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996) xii+253pp., (with a preface by Luc Illusie, edited and with a note by Georges Maltsiniotis. MR1453167 (98c:18007)).  Weibel, C.A., An introduction to homological algebra, Cambridge studies in advanced mathematics, vol. 38, (1994), Cambridge University Press Cambridge, MR1269324 (95f:18001)  Yoneda, N., On ext and exact sequences, J. fac. sci. univ. Tokyo sect. I, 8, 507-576, (1960), MR0225854 (37 #1445) · Zbl 0163.26902
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