Symmetric Monoidal Categories for Operads - Index of

204 13 The (Co)homology of Algebras over Operads
coefficients for algebras in dg-modules is already defined in [26] by methods
of homotopical algebra, the universal coefficient spectral sequences are defined
in [1] for algebras in dg-modules (in the case of non-negatively graded dgmodules).
Throughout this chapter, we use the short notation UP(A) =UP(A)(1) to
refer to the enveloping algebra of P-algebras.
13.1 The Construction
The definition of the cohomology and homology of algebras over operads
makes sense for operads and algebras in simplicial modules, or in dg-modules.
In this section, we give a summary of the constructions in the dg-module
setting. We take C =dgk Mod as a base category and E =dgk Mod as an
underlying category of algebras. The principle of generalized point-tensors
can be used to extend the construction to algebras in Σ∗-objects, and to
algebras in right modules over operads.
We fix an operad P. WerequirethatP is Σ∗-cofibrant to ensure that Palgebras
form a semi-model category.
13.1.1 Derivations and Cohomology. Recall that a map θ : A → E,
where A is a P-algebra and E is a representation of A, forms a derivation if
we have the derivation relation
n�
θ(p(a1,...,an)) = p(a1,...,θ(ai),...,an)
i=1
for all operations p ∈ P(n) andalla1,...,an ∈ A. In the dg-context, we
extend this definition to homogeneous maps θ : A → E to obtain a dg-module
of derivations.
The cohomology of A with coefficients in E is defined by:
H ∗ P (A, E) =H ∗ (DerP(QA,E)),
where QA is any cofibrant replacement of A in the category of P-algebras. In
∼
this definition, we use the augmentation of the cofibrant replacement QA −→
A to equip E with the structure of a representation of QA.
More generally, for any P-algebra B over A, we can use the restriction of
structures through the augmentation ɛ : B → A to equip E with the structure
of a representation of B. Accordingly, we have a well-defined functor from
the category of P-algebras over A to the category of dg-modules defined by
the map DerP(−,E):B ↦→ DerP(B,E).
The next proposition ensures that the definition of the cohomology makes
sense: