Symmetric Monoidal Categories for Operads - Index of

8.1 The Explicit Construction of the Adjoint Functor ΓR : F R →MR
Our first goal is to check:
8.1.2 Proposition. We have a natural adjunction relation
MorE(SR(M,A),Y) � MorM R (M,End A,Y )
for every M ∈MR, A ∈ RE and Y ∈E.
Proof. By definition, the object SR(M,A) is defined by a reflexive coequalizer
s0
S(M ◦ R,A) d0
d1
S(M,A)
SR(M,A) ,
where d0 is induced by the right R-action on M and d1 is induced by the
left R-action on A. As a consequence, we obtain that the morphism set
MorE(SR(M,A),Y) is determined by an equalizer of the form
MorE(SR(M,A),Y) MorE(S(M,A),Y)
By proposition 2.3.2, we have adjunction relations
MorE(S(M,A),Y) � MorM(M,End A,Y )
d 0
d 1
123
MorE(S(M ◦ R,A),Y) .
and MorE(S(M ◦ R,A),Y) � MorM(M ◦ R, End A,Y )
that transport this equalizer to an isomorphic equalizer of the form:
ker(d 0 ,d 1 ) MorM(M,End A,Y )
d 0
d 1
MorM(M ◦ R, End A,Y ) .
By functoriality, we obtain that the map d 0 in this equalizer is induced
by the morphism ρ : M ◦ R → M that determines the right R-action on M.
Thus we have d 0 (f) =f · ρ, for all f : M → End A,Y ,whereρ denotes the
right R-action on M. On the other hand, by going through the construction of
proposition 2.3.2, we check readily that d 1 (f) =ρ·f ◦R, where in this formula
ρ denotes the right R-action on End A,Y . Hence we have f ∈ ker(d 0 ,d 1 )ifand
only if f : M → End A,Y forms a morphism of right R-modules.
To conclude, we obtain that the adjunction relation of proposition 2.3.2
restricts to an isomorphism
MorE(SR(M,A),Y) � MorM R (M,End A,Y )
and this achieves the proof of proposition 8.1.2. ⊓⊔
8.1.3 Remark. In §2.3.3, we mention that the endomorphism module
End X,Y forms also a left module over End Y , the endomorphism operad of
Y . Explicitly, we have natural composition products