symmetric operad




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 nn-ary operations for all nn \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 Σ n\Sigma_n on the set (or object in the enriching category) of nn-ary operations. A homomorphism of symmetric operads is then a morphism of planar operads that in addition 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 nn-ary operations! Rather, only the fixed points in the nn-ary operations of the Σ n\Sigma_n-action are symmetric operations. If Σ n\Sigma_n acts freely, then the corresponding nn-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 Σ n\Sigma_n-action, while Assoc has Σ n\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.

Structures on the category of symmetric operads

Boardman-Vogt tensor product

The category of symmetric operads becomes a closed symmetric monoidal category for the Boardman-Vogt tensor product.

Model structure

For VV a suitable monoidal model category, the category of VV-enriched symmetric operads carries a good model structure on operads. See there for more details.


Relation to categories


Every locally small category CC may be regarded as a coloured symmetric operation j !(C)j_!(C) over set, with the objects of CC and coulours, and with only unary operation, these being the morphisms in the category

j !(C)(c 1,,c n;c)={C(c 1,c) ifn=1 otherwise. j_!(C)(c_1, \cdots, c_n ; c) = \left\{ \array{ C(c_1, c) & if \, n = 1 \\ \emptyset & otherwise } \right. \,.

This functor j !:CatOperadj_! : Cat \to Operad has a right adjoint j *:OperadCatj^* : Operad \to Cat which sends an operad to the underlying category obtained by discarding all nn-ary operations for n1n \neq 1.

There is a natural isomorphism j*j !idj * j_! \simeq id.

By the discussion at adjoint functor this exhibits a coreflective subcategory

Catj *j !Operad. Cat \stackrel{\overset{j_!}{\hookrightarrow}}{\underset{j^*}{\leftarrow}} Operad \,.

Let η\eta denote the symmetric operad with a single colour and no non-identity operation. Then the slice category of Operad over η\eta is equivalent to Cat

CatOperad /η. Cat \simeq Operad_{/\eta} \,.

Because a morphism of operads PηP \to \eta can exists precisely if there are no operations of arity other than 1 in PP.

Under this identification the fuctor j !j_! is the canonical projection out of the slice category

j !:CatOperad /ηOperad. j_! : Cat \stackrel{\simeq}{\to} Operad_{/\eta} \to Operad \,.

For more on this see at dendroidal set the section The full diagram of relations.

Relation to planar operads

There is an evident forgetful functor

U:SymmetricOperadPlanarOperad U : SymmetricOperad \to PlanarOperad

to the category of planar operads, which forgets the action of the symmetric groups. This functor has a left adjoint

Symm:PlanarOperadSymmetricOperad. Symm : PlanarOperad \to SymmetricOperad \,.

The free construction 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)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 Σ n\Sigma_n in degree nn. This is no longer the terminal object in SymmetricOperadSymmetricOperad, which instead is Comm.


In SetSet

We list some examples of Set-enriched symmetric operads.

  • For every symmetric monoidal category CC, there is naturally the symmetric endomorphism operad End(C)End(C).

    This establishes a reflective (but non-full) inclusion

    End:SymmMonCatSymmOperad End : SymmMonCat \to SymmOperad

    and makes precise the way in which a (symmetric) operad is a generalization of a (symmetric) monoidal category.

    For any other symmetric operad PP, a morphism of symmetric operads

    PEnd(C) P \to End(C)

    is precisely an algebra over an operad over PP in CC.

  • The operad Comm for commutative monoids is the terminal object in symmetric VV-operads, for instance for V=V = Set, sSet, Top, etc.

    It has a single nn-ary operation for all nn \in \mathbb{N}, with the symmetric group necessarily acting trivially in each degree.

    A morphism of operads

    A:CommEnd(Vect) A : Comm \to End(Vect)

    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!n! many operations in degree nn, with the symmetric group acting freely on these.

    This means that there is a single nn-ary operation “up to a choice of ordering of its arguments”.

    A morphism of operads

    AssocEnd(Vect) Assoc \to End(Vect)

    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.

In TopTop



An original source is

  • Peter May, The geometry of iterated loop spaces, Lectures Notes in Mathematics, Vol. 271, Springer-Verlag, Berlin-New York, (1972).

A survey of the basic notions of symmetric operads is for instance in section 1 of

See the references at operad for more.

Expression of symmetric operads as polynomial 2-monads is discussed in

  • 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)

  • Mark Weber, Operads as polynomial 2-monads (arXiv:1412.7599)

Last revised on January 11, 2019 at 01:30:10. See the history of this page for a list of all contributions to it.