symmetric monoidal (∞,1)-category of spectra
The notion of operad comes in two broad flavors (apart from the choice of enriching category): symmetric operads and planar operads.
Roughly, a planar operad consists of $n$-ary operations for all $n \in \mathbb{N}$ equipped with a suitable notion of compositon, while a symmetric operad in addition is equipped with a compatible action of the symmetric group $\Sigma_n$ on the set (or object in the enriching category) of $n$-ary operations. A homomorphism of symmetric operads is then a morphism of planar operads that in additions respects this action.
The extra “symmetry” structure carried by symmetric operads is crucial for the behaviour of the category of operads in many applications (see the examples below). Notice that it does not so much affect the idea of what a single operad is. In particular, symmetric operads are not restricted to encoding algebraic symmetric structures with symmetric $n$-ary operations! Rather, only the fixed points in the $n$-ary operations of the $\Sigma_n$-action are symmetric operations. If $\Sigma_n$ acts freely, then the corresponding $n$-ary operations are still maximally non-symmetric themselves.
The central example illustrating this are the operads Comm and Assoc. Regarded as symmetric Set-enriched operads, Comm has the singleton set in each degree, with trivial $\Sigma_n$-action, while Assoc has $\Sigma_n$ in each degree, freely acting on itself.
Therefore Comm is the terminal object in the category of symmetric operads (while Assoc, regarded as a planar operad, is the terminal object in that category).
Multi-coured symmetric operads are equivalently known also as symmetric multicategories.
The category of symmetric operads becomes a closed symmetric monoidal category for the Boardman-Vogt tensor product.
For $V$ a suitable monoidal model category, the category of $V$-enriched symmetric operads carries a good model structure on operads. See there for more details.
Every locally small category $C$ may be regarded as a coloured symmetric operation $j_!(C)$ over set, with the objects of $C$ and coulours, and with only unary operation, these being the morphisms in the category
This functor $j_! : Cat \to Operad$ has a right adjoint $j^* : Operad \to Cat$ which sends an operad to the underlying category obtained by discarding all $n$-ary operations for $n \neq 1$.
There is a natural isomorphism $j * j_! \simeq id$.
By the discussion at adjoint functor this exhibits a coreflective subcategory
Let $\eta$ denote the symmetric operad with a single colour and no non-identity operation. Then the slice category of Operad over $\eta$ is equivalent to Cat
Because a morphism of operads $P \to \eta$ can exists precisely if there are no operations of arity other than 1 in $P$.
Under this identification the fuctor $j_!$ is the canonical projection out of the slice category
For more on this see at dendroidal set the section The full diagram of relations.
There is an evident forgetful functor
to the category of planar operads, which forgets the action of the symmetric groups. This functor has a left adjoint
The freely adds symmetric group actions.
For instance as a planar operad, Assoc is the terminal object (has the point in each degree). Its symmetrization $Symm(Assoc)$ is still the operad for associative monoids, now regarded as a symmetric operad, where it has the underlying set of the symmetric group $\Sigma_n$ in degree $n$. This is no longer the terminal object in $SymmetricOperad$, which instead is Comm.
We list some examples of Set-enriched symmetric operads.
For every symmetric monoidal category $C$, there is naturally the symmetric endomorphism operad $End(C)$.
This establishes a reflective (but non-full) inclusion
and makes precise the way in which a (symmetric) operad is a generalization of a (symmetric) monoidal category.
For any other symmetric operad $P$, a morphism of symmetric operads
is precisely an algebra over an operad over $P$ in $C$.
The operad Comm for commutative monoids is the terminal object in symmetric $V$-operads, for instance for $V =$ Set, sSet, Top, etc.
It has a single $n$-ary operation for all $n \in \mathbb{N}$, with the symmetric group necessarily acting trivially in each degree.
A morphism of operads
is precisely a commutative and associative algebra structure on a vector space.
The operad Assoc for monoids is, as a symmetric operad, the one with a single colour that has precisely $n!$ many operations in degree $n$, with the symmetric group acting freely on these.
This means that there is a single $n$-ary operation “up to a choice of ordering of its arguments”.
A morphism of operads
is precisely an associative algebra on a vector space.
For every non-planar finite rooted tree there is a symmetric operad freely generated by it. For more on this see the section Trees and free operads at dendroidal set.
(…)
An original source is
A survey of the basic notions of symmetric operads is for instance in section 1 of
See the references at operad for more.