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1 results

  • 7 - Derived Functors and Abelian Monoidal Model Structures

  • James Gillespie, Ramapo College of New Jersey
  • Book:
    Abelian Model Category Theory
    Published online:
    19 December 2024
    Print publication:
    02 January 2025, pp 177-207

  • Summary

    This chapter studies the left and right derived functors of an additive functor whose domain is a (closed) abelian model category. The most important scenario is when we have a Quillen adjunction, and we show that its left and right derived functors induce a triangulated adjunction of homotopy categories. We characterize Quillen equivalences, which are the Quillen adjunctions that induce triangle equivalences of homotopy categories. The end of the chapter turns to the basics of abelian monoidal model structures. The main result is that a tensor product on the ground category, which is compatible with the model structure, will descend to a well-behaved tensor product on the homotopy category. We give Hovey’s criteria for abelian monoidal model structures which provides a powerful way to construct tensor triangulated categories.