Symmetric Monoidal Categories for Operads - Index of

140 10 Miscellaneous Examples
10.1.1 Shifted Operads. Recall that the shifted operad R[ · ] consists of the
collection of objects R[m](n) =R(m + n), m, n ∈ N.
In §4.1.4, we use the action of Σn on {m +1,...,m+ n} ⊂{1,...,m+ n}
to define an action of Σn on R(m + n), for every n ∈ N, and to give to the
collection R[m](n), n ∈ N, the structure of a Σ∗-object. In the case of an
operad, the composites at last positions
R(r + s) ⊗ R(n1) ⊗···⊗R(ns) → R(r + n1 + ···+ ns)
(insert operad units η : 1 → R(1)atpositions1,...,r) provide each object
R[r], r ∈ N, with the additional structure of a right R-module.
In §4.1.5, we also observe that the collection {R[r]}r∈N is equipped with
the structure of an operad: the action of Σm on {1,...,m}⊂{1,...,m+ n}
determines an action of Σm on R[m](n) =R(m + n), for every m, n ∈ N, and
we use operadic composites at first positions
R(r + s) ⊗ R(m1 + n1) ⊗···⊗R(mr + nr) → R(m1 + n1 + ···+ mr + nr + s)
together with the action of appropriate bloc permutations
R(m1 + n1 + ···+ mr + nr + s) � −→ R(m1 + ···+ mr + n1 + ···+ nr + s)
to define composition products:
R[r] ⊗ R[m1] ⊗···⊗R[mr] → R[m1 + ···+ mr].
The axioms of operads (in May’s form) imply immediately that the action of
permutations w ∈ Σm preserves the internal right R-module structure of R[m]
and similarly regarding the composition products of R[ · ]. Finally, we obtain
that the collection {R[r]}r∈N forms an operad in the symmetric monoidal
category of right R-modules.
The morphism of §4.1.6
η : R → R[ · ],
which identifies R(m) with the constant part of R[m], forms a morphism of
operads in right R-modules.
In the sequel, we use the notation OR to refer to the category of operads
in right R-modules, and the notation R / OR to refer to the comma category
of objects under R ∈O, where we identify the objects P(m) underlying an
operad P ∈Owith constant right R-modules to form a functor O→OR. Our
definition makes the operad R[ · ] an object of this category R/OR.
10.1.2 Functors on Operads. In §4.1.5, we use that S : M ↦→ S(M) defines
a functor of symmetric monoidal categories S : M→F to obtain that the
collection {S(R[r],X)}r∈N, associated to any object X ∈E, forms an operad
in E. We obtain similarly that the collection {SR(R[r],A)}r∈N, associatedto
any R-algebra A ∈ RE, forms an operad in E.