Symmetric Monoidal Categories for Operads - Index of

Chapter 8
Adjunction and Embedding Properties
Introduction
Let R be an operad in C. In this chapter, we generalize results of §2.3 to the
functor SR : M R →FR from the category of right R-modules to the category
of functors F : RE → E,whereE is a fixed symmetric monoidal category
over C.
Again, since we observe in §5.2 that the functor SR : M R →FR preserves
colimits, we obtain that this functor has a right adjoint ΓR : F R →MR.
In §8.1, we generalize the construction of §2.3.4 to give an explicit definition
of this functor ΓR : G ↦→ ΓR(G).
In §8.2, we prove that the functor SR : M R ↦→ FR is faithful in an enriched
sense(likeS:M ↦→ F) if the category E is equipped with a faithful functor
η : C→E. Equivalently, we obtain that the adjunction unit η(M) :M →
ΓR(SR(M)) forms a monomorphism.
In the case E = C = k Mod, we observe that the unit η(M) :M → Γ(S(M))
of the adjunction S : M ⇄ F : Γ forms an isomorphism if M is a projective
Σ∗-module or if the ground ring is an infinite field. In §8.3, we extend these
results to the context of right R-modules:
Theorem 8.A. In the case E = C = k Mod, the adjunction unit ηR(M) :
M → ΓR(SR(M)) defines an isomorphism if M is a projective Σ∗-module or
if the ground ring is an infinite field.
As a corollary, we obtain that the functor SR : M R →F R is full and
faithful in the case E = C = k Mod, where k is an infinite field.
To prove this theorem, we observe that the underlying Σ∗-object of ΓR(G),
forafunctorG : RE → E, is identified with the Σ∗-object Γ(G ◦ R(−))
associated to the composite of G : RE →Ewith the free R-algebra functor
R(−) :E→ RE. ThenweusetherelationSR(M) ◦ R(−) � S(M), for a functor
of the form G = SR(M), to deduce theorem 8.A from the corresponding
assertions about the unit of the adjunction S : M ⇄ F :Γ.
B. Fresse, Modules over Operads and Functors, Lecture Notes in Mathematics 1967, 121
DOI: 10.1007/978-3-540-89056-0 8, c○ Springer-Verlag Berlin Heidelberg 2009