Reedy Diagrams in V-Model Categories

  • Moncef GhazelEmail author
  • Fethi Kadhi


We study the category of Reedy diagrams in a \(\mathscr {V}\)-model category. Explicitly, we show that if K is a small category, \(\mathscr {V}\) is a closed symmetric monoidal category and \(\mathscr {C}\) is a closed \(\mathscr {V}\)-module, then the diagram category \(\mathscr {V}^K\) is a closed symmetric monoidal category and the diagram category \(\mathscr {C}^K\) is a closed \(\mathscr {V}^K\)-module. We then prove that if further K is a Reedy category, \(\mathscr {V}\) is a monoidal model category and \(\mathscr {C}\) is a \(\mathscr {V}\)-model category, then with the Reedy model category structures, \(\mathscr {V}^K\) is a monoidal model category and \(\mathscr {C}^K\) is a \(\mathscr {V}^K\)-model category provided that either the unit 1 of \(\mathscr {V}\) is cofibrant or \(\mathscr {V}\) is cofibrantly generated.


Quillen model category Reedy model structure Symmetric monoidal category Module over a symmetric monoidal model category 

Mathematics Subject Classification

55U35 18D10 19D23 18D15 


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We would like to thank the editor and the reviewer for their thoughtful ideas and constructive comments. Their suggestions substantially improved the quality of the manuscript.


  1. 1.
    Barwick, C.: On left and right model categories and left and right Bousfield localizations. Homol. Homotopy. Appl. 12, 245–320 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berger, C., Moerdijk, I.: On an extension of the notion of Reedy category. Math. Z. 269(34), 9771004 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Borceux, F.: Handbook of Categorical Algebra 2, Categories and Structures, Encyclopedia of Mathematics and its Applications, vol. 51. Cambridge University Press, Cambridge (1994)CrossRefzbMATHGoogle Scholar
  4. 4.
    Day, B.J.: On Closed Categories of Functors. Lecture Notes in Mathematics, 304. Springer, Berlin (1972)zbMATHGoogle Scholar
  5. 5.
    Dwyer, W.G., Hirschhorn, P.S., Kan, D.M.: Model categories and more general abstract homotopy theory: a work in what we like to think of as progress (preprint)Google Scholar
  6. 6.
    Elmendorf, A.D., Kriz, I., Mandell, M.A., May, J.P.: Rings, Modules and Algebras in Stable Homotopy Theory. Surveys Monographs, vol. 47. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  7. 7.
    Goerss, P., Jardine, J.: Simplicial Homotopy Theory. Progress in Mathematics, 174. Birkhäuser, Basel (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Goerss, P., Schemmerhorn, K.: Model categories and simplicial methods. arXiv:math/0609537v2 [math.AT] (28 Nov 2006)
  9. 9.
    Hirschhorn, P.S.: Model Categories and Their Localizations, vol. 99. Surveys and Monographs of the American Mathematical Society, Providence (2003)zbMATHGoogle Scholar
  10. 10.
    Hovey, M.: Monoidal model categories. arXiv:math/9803002v1 [math.AT] (28 Feb 1998)
  11. 11.
    Hovey, M.: Model Categories, vol. 63. Surveys and Monographs of the American Mathematical Society, Providence (1999)zbMATHGoogle Scholar
  12. 12.
    Hovey, M., Shipley, B., Smith, J.: Symmetric spectra. J. Am. Math. Soc. 13(1), 149208 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Janelidze, G., Kelly, G.M.: A note on actions of a monoidal category. Theory Appl. Categ. 9, 61–91 (2001)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kelly, G.M.: Basic concepts of enriched category theory. Repr. Theory Appl. Categ. 10, 1–136 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5, 2nd edn. Springer, New York (1998)zbMATHGoogle Scholar
  16. 16.
    Quillen, D.: Homotopical Algebra. Lecture Notes in Mathemetics 43. Springer, Berlin (1967)CrossRefzbMATHGoogle Scholar
  17. 17.
    Riehl, E.: Categorical Homotopy Theory. New Mathematical Monographs, vol. 24. Cambridge University Press, Cambridge (2014)CrossRefzbMATHGoogle Scholar
  18. 18.
    Riehl, E., Verity, D.: The theory and practice of Reedy categories. Theory Appl. Categ. 29, 256–301 (2014)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Schwede, S., Shipley, B.: Algebras and modules in monoidal model categories. In: Proceedings of London Mathematical Society, vol. 80 (2000)Google Scholar
  20. 20.
    Schwede, S., Shipley, B.: Stable model categories are categories of modules. Topology 42(1), 103–153 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    White, D.: Model structures on commutative monoids in general model categories. J. Pure Appl. Algebra 221, 3124–3168 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Faculté des Sciences Mathématiques, Physiques et Naturelles de TunisUniversity of Tunis El ManarEl Manar TunisTunisia
  2. 2.Ecole Nationale des Sciences de l’InformatiqueManouba UniversityManoubaTunisia

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