A note on model structures on arbitrary Frobenius categories
- First Online:
- 57 Downloads
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).
KeywordsFrobenius categorie triangulated categories model structure
MSC 201018E10 18E30 18E35
Unable to display preview. Download preview PDF.
- P. Gabriel, M. Zisman: Calculus of Fractions and Homotopy Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer, New York, 1967.Google Scholar
- 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
- P. S. Hirschhorn: Model Categories and Their Localizations. Mathematical Surveys and Monographs 99, American Mathematical Society, Providence, 2003.Google Scholar
- M. Hovey: Model Categories. Mathematical Surveys and Monographs 63, American Mathematical Society, Providence, 1999.Google Scholar
- S. MacLane: Categories for the Working Mathematician. Graduate Texts in Mathematics 5, Springer, New York, 1998.Google Scholar
- D. G. Quillen: Homotopical Algebra. Lecture Notes in Mathematics 43, Springer, Berlin, 1967.Google Scholar
- 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