Symmetric Monoidal Categories for Operads - Index of

Chapter 16
Extension and Restriction Functors
and Model Structures
Introduction
In §3.3.5, we recall that an operad morphism φ : P → Q induces adjoint
extension and restriction functors
φ! : PE ⇄ QE : φ ∗
In §7.2, we observe that an operad morphism ψ : R → S induces similar
adjoint extension and restriction functors on module categories:
ψ! : M R ⇄ M S : ψ ∗
In this chapter, we study the functors on model categories defined by these
extension and restriction functors. Our goal is to prove:
Theorem 16.A. Let φ : P → Q be an operad morphism. Suppose that the operad
P (respectively, Q) isΣ∗-cofibrant and use proposition 12.3.A to equip the
category of P-algebras (respectively, Q-algebras) with a semi model structure.
The extension and restriction functors
φ! : PE ⇄ QE : φ ∗
define Quillen adjoint functors. If φ : P → Q is a weak-equivalence, then these
functors define Quillen adjoint equivalences.
Theorem 16.B. Let ψ : R → S be an operad morphism. Assume that the
operad R (respectively, S) isC-cofibrant and use proposition 14.1.A to equip
the category of right R-modules (respectively, right S-modules) with a model
structure.
The extension and restriction functors
ψ! : M R ⇄ M S : ψ ∗
B. Fresse, Modules over Operads and Functors, Lecture Notes in Mathematics 1967, 235
DOI: 10.1007/978-3-540-89056-0 16, c○ Springer-Verlag Berlin Heidelberg 2009