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Reedy Diagrams in V-Model Categories

  • Moncef GhazelEmail author
  • Fethi Kadhi
Article

Abstract

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.

Keywords

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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Notes

Acknowledgements

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.

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Copyright information

© Springer Nature B.V. 2019

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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