Symmetric Monoidal Categories for Operads - Index of

10 Introduction
study the homotopy invariance of the functor S(M) :E→Eassociated to a
Σ∗-object M. We prove more globally that the bifunctor (M,X) ↦→ S(M,X)
satisfies an analogue of the pushout-product axiom of tensor products in
symmetric monoidal model categories.
§12. The Homotopy of Algebras over Operads
To have a good homotopy theory for algebras over an operad P, wehave
to make assumptions on the operad P. In general, it is only reasonable to
assume that the unit morphism of the operad η : I → P defines a cofibration
in the underlying category of Σ∗-objects – we say that the operad P is Σ∗cofibrant.
In the differential graded context, this assumption implies that
each component of the operad P(n) forms a chain complex of projective Σnmodules.
For certain good symmetric monoidal categories, we can simply
assume that the unit morphism of the operad η : I → P defines a cofibration
in the base category C – we say that the operad P is C-cofibrant. In the
differential graded context, this assumption implies that the chain complex
P(n) consists of projective k-modules, but we do not assume that P(n) forms
a chain complex of projective Σn-modules.
The category of P-algebras PE inherits natural classes of weak-equivalences,
cofibrations, fibrations, which are defined by using the adjunction between
PE and the underlying category E. But none of our assumptions, though
reasonable, is sufficient to imply the full axioms of model categories: the lifting
and cofibration axioms hold in the category of P-algebras for morphisms with
a cofibrant domain only. In this situation, it is usual to say that P-algebras
forms a semi-model category.
In fact, the structure of a semi-model category is sufficient to carry out
every usual construction of homotopical algebra. For our purpose, we review
the definition of model structures by adjunction and the notion of a Quillen
adjunction in the context of semi-model categories.
We prove the existence of semi-model structures on the category of Palgebras
PE for every symmetric monoidal model category E over the base
category C when the operad P is Σ∗-cofibrant. Since we prove that Σ∗-objects
form a symmetric monoidal model category over C, we can apply our result
to obtain a semi-model structure on the category of P-algebras in Σ∗-objects,
equivalently, on the category of left P-modules. Recall that a Σ∗-object is
connected if M(0) = 0. We observe that the semi-model structure of the
category of P-algebras in connected Σ∗-objects is well defined as long as the
operad P is C-cofibrant.
§13. The (Co)homology of Algebras over Operads
The category of algebras over an operad P has a natural cohomology theory
H∗ P (A, E) defined as a derived functor from the functor of derivations