Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 329–337

A note on model structures on arbitrary Frobenius categories



We show that there is a model structure in the sense of Quillen on an arbitrary Frobenius category F such that the homotopy category of this model structure is equivalent to the stable category 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 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).


Frobenius categorie triangulated categories model structure 

MSC 2010

18E10 18E30 18E35 


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  1. [1]
    H. Becker: Models for singularity categories. Adv. Math. 254 (2014), 187–232.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    A. Beligiannis, I. Reiten: Homological and homotopical aspects of torsion theories. Mem. Am. Math. Soc. 188 (2007), 207 pages.MathSciNetMATHGoogle Scholar
  3. [3]
    T. Bühler: Exact categories. Expo. Math. 28 (2010), 1–69.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    W. G. Dwyer, J. Spalinski: Homotopy theories and model categories, Handbook of Algebraic Topology. North-Holland, Amsterdam, 1995, pp. 73–126.MATHGoogle Scholar
  5. [5]
    P. Gabriel, M. Zisman: Calculus of Fractions and Homotopy Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer, New York, 1967.Google Scholar
  6. [6]
    J. Gillespie: Model structures on exact categories. J. Pure Appl. Algebra 215 (2011), 2892–2902.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    J. Gillespie: Exact model structures and recollements. J. Algebra 458 (2016), 265–306.MathSciNetCrossRefMATHGoogle Scholar
  8. [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.Google Scholar
  9. [9]
    P. S. Hirschhorn: Model Categories and Their Localizations. Mathematical Surveys and Monographs 99, American Mathematical Society, Providence, 2003.Google Scholar
  10. [10]
    M. Hovey: Model Categories. Mathematical Surveys and Monographs 63, American Mathematical Society, Providence, 1999.Google Scholar
  11. [11]
    M. Hovey: Cotorsion pairs, model category structures, and representation theory. Math. Z. 241 (2002), 553–592.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    B. Keller: Chain complexes and stable categories. Manuscr. Math. 67 (1990), 379–417.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    S. MacLane: Categories for the Working Mathematician. Graduate Texts in Mathematics 5, Springer, New York, 1998.Google Scholar
  14. [14]
    D. G. Quillen: Homotopical Algebra. Lecture Notes in Mathematics 43, Springer, Berlin, 1967.Google Scholar
  15. [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, pp. 85–147.Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhou, JiangsuP.R. China

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