symmetric monoidal (∞,1)-category of spectra
Roughly, a planar operad consists of -ary operations for all equipped with a suitable notion of compositon, while a symmetric operad in addition is equipped with a compatible action of the symmetric group on the set (or object in the enriching category) of -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 -ary operations! Rather, only the fixed points in the -ary operations of the -action are symmetric operations. If acts freely, then the corresponding -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 -action, while Assoc has in each degree, freely acting on itself.
Multi-coured symmetric operads are equivalently known also as symmetric multicategories.
Every locally small category may be regarded as a coloured symmetric operation over set, with the objects of and coulours, and with only unary operation, these being the morphisms in the category
There is a natural isomorphism .
Because a morphism of operads can exists precisely if there are no operations of arity other than 1 in .
Under this identification the fuctor is the canonical projection out of the slice category
There is an evident forgetful functor
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 is still the operad for associative monoids, now regarded as a symmetric operad, where it has the underlying set of the symmetric group in degree . This is no longer the terminal object in , which instead is Comm.
We list some examples of Set-enriched symmetric operads.
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 , a morphism of symmetric operads
is precisely an algebra over an operad over in .
It has a single -ary operation for all , with the symmetric group necessarily acting trivially in each degree.
A morphism of operads
This means that there is a single -ary operation “up to a choice of ordering of its arguments”.
A morphism of operads
is precisely an associative algebra on a vector space.
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.
Joachim Kock, Data types with symmetries and polynomial functors over groupoids, 28th Conference on the Mathematical Foundations of Programming Semantics (Bath, June 2012); in Electronic Notes in Theoretical Computer Science. (arXiv:1210.0828)