Symmetric Monoidal Categories for Operads - Index of

270 18 Shifted Modules over Operads and Functors
in the category of right R-modules. In §19, we use this construction to analyze
the image of relative cell complexes under functors associated to right
R-modules.
The objects LnM[Y/X]andTnM[Y/X] are defined in §§18.2.6-18.2.10.
A preliminary step is to reduce the construction of pushouts to reflexive
coequalizers of free R-algebras, because we have immediately:
18.2.1 Proposition.
(a) For a free R-algebra A
SR[M,R(X)] = S[M,X].
= R(X), we have a natural isomorphism
(b) The functor SR[M,−] : RE →MR preserves the coequalizers which are
reflexive in E.
Proof. Since we have an identity SR[M,A](n) =SR(M[n],A), assertion (a) is
a corollary of assertion (b) of theorem 7.1.1, assertion (b) is a corollary of
proposition 5.2.2. ⊓⊔
In any category, a pushout
S A
T B
is equivalent to a reflexive coequalizer such that:
T ∨ S ∨ A
s0
d0
d1
T ∨ A
Thus, by assertion (b) of proposition 18.2.1, the object SR[M,B] thatweaim
to understand is determined by a reflexive coequalizer of the form:
s0
SR[M,R(Y ) ∨ R(X) ∨ A] d0
d1
B.
SR[M,R(Y ) ∨ A]
SR[M,B] .
The motivation to introduce refined functors SR[M,−] appears in the next
observations:
18.2.2 Lemma. For any Σ∗-object M and any sum of objects X ⊕ Y in E,
we have identities
S[M,X ⊕ Y ] � S[M[X],Y] � S[M,Y ][X].