Symmetric Monoidal Categories for Operads - Index of

114 7 Universal Constructions on Right Modules over Operads
E
S(L)
E
U
RE
SR(L◦R)
in which U : RE →Edenotes the forgetful functor, commutes up to a natural
isomorphism of functors.
(b) For any right R-module M ∈MR, the diagram
E
S(M)
R(−)
E
RE
SR(M)
in which R(−) :E→ RE denotes the free R-algebra functor, commutes up to a
natural isomorphism of functors.
In other words, we have functor isomorphisms SR(L ◦ R) � S(L) ◦ U, for
all free right R-modules M = L ◦ R, and S(M) � SR(M) ◦ R(−), for all
right R-modules M ∈MR. The isomorphisms that give the functor relations
of theorem 7.1.1 are also natural with respect to the symmetric monoidal
category E on which functors are defined.
This theorem occurs as the particular case of an operad unit η : I → R in
theorem 7.2.2 proved next. Note that we prove directly the general case of
theorem 7.2.2 and our arguments do not rely on theorem 7.1.1. Therefore we
can refer to theorem 7.2.2 for the proof of this proposition.
7.1.2 Generating Objects in Right Modules over Operads. Consider
thefreerightR-modules Fr ◦ R, r ∈ N, associated to the generators Fr, r ∈ N
of the category of Σ∗-objects.
As Fr = I⊗r ,wehaveF1◦R = I ◦R � R, the operad R considered as a right
module over itself, and, for all r ∈ N, weobtainFr◦R =(I⊗r ) ◦ R � R⊗r ,the
rth tensor power of R in the category of right R-modules.
Since S(Fr) � Id ⊗r ,therth tensor power of the identity functor Id : E→
E, theorem 7.1.1 returns:
7.1.3 Proposition. We have SR(F1 ◦ R) =SR(R) =U, the forgetful functor
U : RE →E,andSR(Fr◦R) =SR(R) ⊗r = U ⊗r ,therth tensor power of the
forgetful functor U : RE →E. ⊓⊔
The next proposition is an immediate consequence of proposition 2.1.13
and of the adjunction relation HomM R(L ◦ R,M) � HomM(L, M).
7.1.4 Proposition. We have a natural isomorphism
ωr(M) :M(r) � −→ HomM R (Fr ◦ R,M)
for every M ∈MR. ⊓⊔
,
,