Symmetric Monoidal Categories for Operads - Index of

182 11 Symmetric Monoidal Model Categories for Operads
Proof. Similar to proposition 11.5.3. ⊓⊔
In usual cases, we use the next observation to prove that a category E has
regulartensorpowers:
11.6.5 Observation. Suppose the class of B-cofibrations is created by a
forgetful functor U : E→Bwhich preserves colimits, cofibrations, and weakequivalences.
The category E has regular tensor powers with respect to B-cofibrations
if, for every n ∈ N, thenfold pushout-product λ(i) :Ln(Y/X) → Tn(Y/X)
defines a cofibration (respectively, an acyclic cofibration) in the projective
model category of Σn-objects in B whenever i is a B-cofibration (respectively,
an acyclic B-cofibration) in E.
The examples of reduced categories with regular tensor powers introduced
in this book arise from:
11.6.6 Proposition. The category of connected Σ∗-objects M 0 has regular
tensor powers with respect to C-cofibrations.
Proof. Let M ∈M 0 .In[14,§1.3.7-1.3.9] we observe that the object M ⊗r (n)
expands as a sum of tensor products M ⊗r (n) = �
(I1,...,Ir) M(I1)⊗···⊗M(Ir)
where (I1,...,Ir) ranges over partitions I1 ∐···∐Ir = {1,...,n} so that
Ik �= ∅, for every k ∈{1,...,r}. The symmetric group Σr operates on M ⊗r (n)
by permuting the factors of the tensor product N(I1) ⊗···⊗N(Ir) andthe
summands of the expansion. The assumption Ik �= ∅ implies that Σr acts
freely on partitions (I1,...,Ir). As a corollary, we have an isomorphism of
Σr-objects
M ⊗r � �
�
(n) � Σr ⊗ M(I1) ⊗···⊗M(Ir)
(I1,...,Ir) ′
where the sum ranges of representative of the coset of partitions (I1,...,Ir)
under the action of Σr (see loc. cit. for details).
From this assertion, we deduce readily that the category of connected Σ∗objects
M 0 satisfies the assumption of observation 11.6.5. Hence we conclude
that M 0 has regular tensor powers with respect to C-cofibrations. ⊓⊔
11.6.7 Remark: The Example of Simplicial Modules. Proposition
11.5.3 can also be improved in the context of simplicial modules ∗ .
According to [9], the natural extension of any functor F : k Mod → k Mod
to the category of simplicial k-modules maps a weak-equivalence between
cofibrant objects to a weak-equivalence. From this result, it is easy to deduce
that assertion (a) of proposition 11.5.3 holds for every Σ∗-object in simplicial
k-modules M and not only for cofibrant Σ∗-objects.
∗ Observe however that the category of simplicial sets does not satisfies the axiom of
regular tensor powers.