A note on model structures on arbitrary Frobenius categories

Zhi-wei Li

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 2, page 329-337
  • ISSN: 0011-4642

Abstract

top
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).

How to cite

top

Li, Zhi-wei. "A note on model structures on arbitrary Frobenius categories." Czechoslovak Mathematical Journal 67.2 (2017): 329-337. <http://eudml.org/doc/288177>.

@article{Li2017,
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).},
author = {Li, Zhi-wei},
journal = {Czechoslovak Mathematical Journal},
keywords = {Frobenius categorie; triangulated categories; model structure},
language = {eng},
number = {2},
pages = {329-337},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on model structures on arbitrary Frobenius categories},
url = {http://eudml.org/doc/288177},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Li, Zhi-wei
TI - A note on model structures on arbitrary Frobenius categories
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 329
EP - 337
AB - 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).
LA - eng
KW - Frobenius categorie; triangulated categories; model structure
UR - http://eudml.org/doc/288177
ER -

References

top
  1. Becker, H., 10.1016/j.aim.2013.11.016, Adv. Math. 254 (2014), 187-232. (2014) Zbl06284998MR3161097DOI10.1016/j.aim.2013.11.016
  2. Beligiannis, A., Reiten, I., 10.1090/memo/0883, Mem. Am. Math. Soc. 188 (2007), 207 pages. (2007) Zbl1124.18005MR2327478DOI10.1090/memo/0883
  3. Bühler, T., 10.1016/j.exmath.2009.04.004, Expo. Math. 28 (2010), 1-69. (2010) Zbl1192.18007MR2606234DOI10.1016/j.exmath.2009.04.004
  4. Dwyer, W. G., Spalinski, J., 10.1016/B978-044481779-2/50003-1, Handbook of Algebraic Topology North-Holland, Amsterdam (1995), 73-126. (1995) Zbl0869.55018MR1361887DOI10.1016/B978-044481779-2/50003-1
  5. Gabriel, P., Zisman, M., 10.1007/978-3-642-85844-4, Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer, New York (1967). (1967) Zbl0186.56802MR0210125DOI10.1007/978-3-642-85844-4
  6. Gillespie, J., 10.1016/j.jpaa.2011.04.010, J. Pure Appl. Algebra 215 (2011), 2892-2902. (2011) Zbl1315.18019MR2811572DOI10.1016/j.jpaa.2011.04.010
  7. Gillespie, J., 10.1016/j.jalgebra.2016.03.021, J. Algebra 458 (2016), 265-306. (2016) Zbl06588435MR3500779DOI10.1016/j.jalgebra.2016.03.021
  8. Happel, D., 10.1017/CBO9780511629228, London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge (1988). (1988) Zbl0635.16017MR0935124DOI10.1017/CBO9780511629228
  9. Hirschhorn, P. S., Model Categories and Their Localizations, Mathematical Surveys and Monographs 99, American Mathematical Society, Providence (2003). (2003) Zbl1017.55001MR1944041
  10. Hovey, M., Model Categories, Mathematical Surveys and Monographs 63, American Mathematical Society, Providence (1999). (1999) Zbl0909.55001MR1650134
  11. Hovey, M., 10.1007/s00209-002-0431-9, Math. Z. 241 (2002), 553-592. (2002) Zbl1016.55010MR1938704DOI10.1007/s00209-002-0431-9
  12. Keller, B., 10.1007/BF02568439, Manuscr. Math. 67 (1990), 379-417. (1990) Zbl0753.18005MR1052551DOI10.1007/BF02568439
  13. Lane, S. Mac, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer, New York (1998). (1998) Zbl0906.18001MR1712872
  14. Quillen, D. G., 10.1007/BFb0097438, Lecture Notes in Mathematics 43, Springer, Berlin (1967). (1967) Zbl0168.20903MR0223432DOI10.1007/BFb0097438
  15. Quillen, D., 10.1007/BFb0067053, Algebraic -Theory I. Proc. Conf. Battelle Inst. 1972, Lecture Notes in Mathematics 341, Springer, Berlin (1973), 85-147. (1973) Zbl0292.18004MR0338129DOI10.1007/BFb0067053

NotesEmbed ?

top

You must be logged in to post comments.