Symmetric Monoidal Categories for Operads - Index of

Chapter 10
Miscellaneous Examples
Introduction
Many usual functors are defined by right modules over operads. In this
chapter we study the universal constructions of §4: enveloping operads
(§10.1), enveloping algebras (§10.2), and Kähler differentials (§10.3).
In §§10.2-10.3, we apply the principle of generalized point-tensors to extend
constructions of §4 in order to obtain structure results on the modules
which represent the enveloping algebra and Kähler differential functors. The
examples of these sections are intended as illustrations of our constructions.
In §17, we study other constructions which occur in the homotopy theory
of algebras over operads: cofibrant replacements, simplicial resolutions, and
cotangent complexes. New instances of functors associated to right modules
over operads can be derived from these examples by using categorical operations
of §6, §7 and§9.
10.1 Enveloping Operads and Algebras
Recall that the enveloping operad of an algebra A over an operad R is the
object UR(A) of the category of operads under R defined by the adjunction
relation
Mor R / O(UR(A), S) =Mor RE(A, S(0)).
The goal of this section is to prove that the functor A ↦→ UR(A) isassociated
to an operad in right R-modules, the shifted operad R[ · ] introduced in
the construction of §4.1.
In §4.1, we only prove that R[ · ] forms an operad in Σ∗-objects. First, we
review the definition of the shifted operad R[ · ] to check that the components
R[m], m ∈ N, come equipped with the structure of a right R-module so that
R[ · ] forms an operad in that category. Then we observe that the construction
of §4.1 identifies the enveloping operad UR(A) with the functor SR(R[ · ],A)
associated to the shifted operad R[ · ].
B. Fresse, Modules over Operads and Functors, Lecture Notes in Mathematics 1967, 139
DOI: 10.1007/978-3-540-89056-0 10, c○ Springer-Verlag Berlin Heidelberg 2009