Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 329-337, 2017

A note on model structures on arbitrary Frobenius categories

Zhi-Wei Li

Received October 26, 2015. First published March 1, 2017.

Abstract: We show that there is a model structure in the sense of Quillen on an arbitrary Frobenius category $\mathcal{F}$ such that the homotopy category of this model structure is equivalent to the stable category $\underline{\mathcal{F}}$ as triangulated categories. This seems to be well-accepted by experts but we were unable to find a complete proof for it in the literature. When $\mathcal{F}$ is a weakly idempotent complete (i.e., every split monomorphism is an inflation) Frobenius category, the model structure we constructed is an exact (closed) model structure in the sense of Gillespie (2011).

Keywords: Frobenius categorie; triangulated categories; model structure

Classification MSC: 18E10, 18E30, 18E35

DOI: 10.21136/CMJ.2017.0582-15

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

References:
[1] H. Becker: Models for singularity categories. Adv. Math. 254 (2014), 187-232. DOI 10.1016/j.aim.2013.11.016 | MR 3161097 | Zbl 06284998
[2] A. Beligiannis, I. Reiten: Homological and homotopical aspects of torsion theories. Mem. Am. Math. Soc. 188 (2007), 207 pages. DOI 10.1090/memo/0883 | MR 2327478 | Zbl 1124.18005
[3] T. Bühler: Exact categories. Expo. Math. 28 (2010), 1-69. DOI 10.1016/j.exmath.2009.04.004 | MR 2606234 | Zbl 1192.18007
[4] W. G. Dwyer, J. Spalinski: Homotopy theories and model categories. Handbook of Algebraic Topology North-Holland, Amsterdam (1995), 73-126. DOI 10.1016/B978-044481779-2/50003-1 | MR 1361887 | Zbl 0869.55018
[5] P. Gabriel, M. Zisman: Calculus of Fractions and Homotopy Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer, New York (1967). DOI 10.1007/978-3-642-85844-4 | MR 0210125 | Zbl 0186.56802
[6] J. Gillespie: Model structures on exact categories. J. Pure Appl. Algebra 215 (2011), 2892-2902. DOI 10.1016/j.jpaa.2011.04.010 | MR 2811572 | Zbl 1315.18019
[7] J. Gillespie: Exact model structures and recollements. J. Algebra 458 (2016), 265-306. DOI 10.1016/j.jalgebra.2016.03.021 | MR 3500779 | Zbl 06588435
[8] D. Happel: Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras. London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge (1988). DOI 10.1017/CBO9780511629228 | MR 0935124 | Zbl 0635.16017
[9] P. S. Hirschhorn: Model Categories and Their Localizations. Mathematical Surveys and Monographs 99, American Mathematical Society, Providence (2003). MR 1944041 | Zbl 1017.55001
[10] M. Hovey: Model Categories. Mathematical Surveys and Monographs 63, American Mathematical Society, Providence (1999). MR 1650134 | Zbl 0909.55001
[11] M. Hovey: Cotorsion pairs, model category structures, and representation theory. Math. Z. 241 (2002), 553-592. DOI 10.1007/s00209-002-0431-9 | MR 1938704 | Zbl 1016.55010
[12] B. Keller: Chain complexes and stable categories. Manuscr. Math. 67 (1990), 379-417. DOI 10.1007/BF02568439 | MR 1052551 | Zbl 0753.18005
[13] S. Mac Lane: Categories for the Working Mathematician. Graduate Texts in Mathematics 5, Springer, New York (1998). MR 1712872 | Zbl 0906.18001
[14] D. G. Quillen: Homotopical Algebra. Lecture Notes in Mathematics 43, Springer, Berlin (1967). DOI 10.1007/BFb0097438 | MR 0223432 | Zbl 0168.20903
[15] D. Quillen: Higher algebraic $K$-theory. I. Algebraic $K$-Theory I. Proc. Conf. Battelle Inst. 1972, Lecture Notes in Mathematics 341, Springer, Berlin (1973), 85-147. DOI 10.1007/BFb0067053 | MR 0338129 | Zbl 0292.18004

Affiliations: Zhi-Wei Li, School of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou 221116, Jiangsu, P. R. China, e-mail: zhiweili@jsnu.edu.cn

Springer subscribers can access the papers on Springer website.

[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]

PDF available at:

Springer
for subscribers

···

Institute of Mathematics CAS
free access

···

Digital Mathematics Library
free access