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Czechoslovak Mathematical Journal, Vol. 69, No. 4, pp. 1089-1099, 2019

On $n$-exact categories

Said Manjra

Received February 2, 2018.   Published online June 5, 2019.
Abstract:  An $n$-exact category is a pair consisting of an additive category and a class of sequences with $n+2$ terms satisfying certain axioms. We introduce $n$-weakly idempotent complete categories. Then we prove that an additive $n$-weakly idempotent complete category together with the class $\mathcal{C}_n$ of all contractible sequences with $n+2$ terms is an $n$-exact category. Some properties of the class $\mathcal{C}_n$ are also discussed.
Keywords:  $n$-exact category; contractible sequence; idempotent complete category
Classification MSC:  18E99, 18E10
DOI:  10.21136/CMJ.2019.0067-18
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References:
[1] F. Borceux: Handbook of Categorical Algebra. Volume 2: Categories and Structures. Encyclopedia of Mathematics and Its Applications, 50, Cambridge University Press, Cambridge (2008). DOI 10.1017/CBO9780511525865 | MR 1313497 | Zbl 1143.18002
[2] T. Bühler: Exact categories. Expo. Math. 28 (2010), 1-69. DOI 10.1016/j.exmath.2009.04.004 | MR 2606234 | Zbl 1192.18007
[3] P. Freyd: Splitting homotopy idempotents. Proc. Conf. Categor. Algebra, La Jolla 1965 (S. Eilenberg et al., eds.). Springer, Berlin (1966), 173-176. DOI 10.1007/978-3-642-99902-4_6 | MR 0206069 | Zbl 0195.52902
[4] C. Geiss, B. Keller, S. Oppermann: $n$-angulated categories. J. Reine Angew. Math. 675 (2013), 101-120. DOI 10.1515/crelle.2011.177 | MR 3021448 | Zbl 1271.18013
[5] O. Iyama: Auslander correspondence. Adv. Math. 210 (2007), 51-82. DOI 10.1016/j.aim.2006.06.003 | MR 2298820 | Zbl 1115.16006
[6] O. Iyama: Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories. Adv. Math. 210 (2007), 22-50. DOI 10.1016/j.aim.2006.06.002 | MR 2298819 | Zbl 1115.16005
[7] O. Iyama: Cluster tilting for higher Auslander algebras. Adv. Math. 226 (2011), 1-61. DOI 10.1016/j.aim.2010.03.004 | MR 2735750 | Zbl 1233.16014
[8] O. Iyama, Y. Yoshino: Mutation in triangulated categories and rigid Cohen-Macaulay modules. Invent. Math. 172 (2008), 117-168. DOI 10.1007/s00222-007-0096-4 | MR 2385669 | Zbl 1140.18007
[9] G. Jasso: $n$-Abelian and $n$-exact categories. Math. Z. 283 (2016), 703-759. DOI 10.1007/s00209-016-1619-8 | MR 3519980 | Zbl 1356.18005
[10] Z. Lin: Right $n$-angulated categories arising from covariantly finite subcategories. Commun. Algebra 45 (2017), 828-840. DOI 10.1080/00927872.2016.1175591 | MR 3562541 | Zbl 1371.18010
[11] A. Neeman: The derived category of an exact category. J. Algebra 135 (1990), 388-394. DOI 10.1016/0021-8693(90)90296-z | MR 1080854 | Zbl 0753.18004
[12] D. Quillen: Higher algebraic $K$-theory. I. Algebraic $K$-Theory I: Higher $K$-theories H. Bass Lecture Notes in Mathematics 341, Springer, Berlin (1973), 85-147. DOI 10.1007/bfb0067053 | MR 0338129 | Zbl 0292.18004
[13] R. W. Thomason, T. Trobaugh: Higher algebraic $K$-theory of schemes and of derived categories. The Grothendieck Festschrift, Vol. III (P. Cartier et al., eds.). Progress in Mathematics 88, Birkhäuser, Boston (1990), 247-435. DOI 10.1007/978-0-8176-4576-2_10 | MR 1106918 | Zbl 0731.14001
Affiliations:   Said Manjra, Department of Mathematics & Statistics, College of Science, IMAM University, P.O.Box: 12068, Riyadh 11473, Saudi Arabia, e-mail: smamanjra@imamu.edu.sa, smanjra@uottawa.ca
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