symmetric sequence




A symmetric sequence is a sequence of objects where the nnth object has an action of the nnth symmetric group.

A symmetric sequence is a species by another name. Meaning: they are categorically equivalent notions.


  1. Let CC be a category and GG a group. A GG-representation of CC is a functor {G}C\bullet\{G\} \to C, where {G}\bullet\{G\} is the category with a single object \bullet and Hom(,)=G\Hom(\bullet, \bullet) = G. Explicitly, a GG-representation of CC is the data of an object XCX \in C together with an action a:GAut C(X)a : G \to \Aut_C(X). We write Rep(G,C)Rep(G, C) for the category of GG-representations of CC.

  2. Let Φ\Phi be a graded monoid in the category of groups. Explicitly this is the data of groups Φ n\Phi_n for all nNn \in \mathbf{N} with morphisms Φ m×Φ nΦ m+n\Phi_m \times \Phi_n \to \Phi_{m+n} for all m,n0m,n \ge 0 (subject to various axioms…). Φ\Phi is usually either Σ=(Σ n) n\Sigma = (\Sigma_n)_n, the graded monoid of symmetric groups, or 1=(1 n) n1 = (1_n)_n, the graded monoid of trivial groups.

  3. A Φ\Phi-symmetric sequence in CC is a sequence of Φ n\Phi_n-representations for n0n \ge 0:

    Seq(Φ,C)= n0Rep(Φ n,C) Seq(\Phi, C) = \sqcup_{n \ge 0} Rep(\Phi_n, C)

    In other words a Φ\Phi-symmetric sequence is a sequence of objects (X n) n0(X_n)_{n \ge 0} together with actions a n:Φ nAut C(X n)a_n : \Phi_n \to \Aut_C(X_n). When Φ=Σ\Phi = \Sigma, the graded monoid of symmetric groups, we say simply “symmetric sequence”.

In the case that the graded group of interest is indeed Σ\Sigma, we can define a Σ\Sigma-symmetric sequence somewhat more simply:


A Σ\Sigma-symmetric sequence in a symmetric monoidal category CC is a functor from FinSetFinSet, the category of finite sets and bijections, to CC.

The relationship between the two definitions is that given a functor F:FinSetCF:FinSet\to C, we have a sequence of objects for each nn associated to the finite set with nn elements. The action of Σ n\Sigma_n on these objects comes from the fact that for every permutation in Σ n\Sigma_n there is an associated morphism in FinSetFinSet. Sometimes the category FinSetFinSet is replaced with its skeleton, the category of all finite ordinals with all bijections between them. This latter category is sometimes denoted FinOrdFinOrd or just Σ\Sigma. In the latter case, we sometimes call Σ\Sigma-symmetric sequences just “Σ\Sigma-sequences.”

Symmetric monoidal structure

  1. Let α:HG\alpha : H \to G be a homomorphism of groups and consider the restriction of scalars functor

    Res H G:Rep(G,C)Rep(H,C) \Res^G_H : \Rep(G, C) \to \Rep(H, C)

    which is defined in the obvious way. It admits a left adjoint

    Ind H G:Rep(H,C)Rep(G,C) Ind^G_H : \Rep(H, C) \to \Rep(G, C)

    called the induced representation functor.

  2. Suppose now that CC has a symmetric monoidal structure. Assume also that CC admits coproducts and that the functors XX \otimes - commute with them (for all XCX \in C). Then there is an induced symmetric monoidal structure on Seq(Φ,C)Seq(\Phi, C). Given symmetric sequences X=(X n) nX = (X_n)_n and Y=(Y n) nY = (Y_n)_n, we define XYX \otimes Y as the symmetric sequence which has in the nnth component

    (XY) n= p+q=nInd Φ p×Φ q Φ n(X pY q) (X \otimes Y)_n = \sqcup_{p+q=n} \Ind^{\Phi_n}_{\Phi_p \times \Phi_q} (X_p \otimes Y_q)

    where Φ p×Φ qΦ n\Phi_p \times \Phi_q \to \Phi_n is the canonical morphism which is part of the structure of the graded monoid Φ=(Φ n) n\Phi = (\Phi_n)_n. The unit with respect to this monoidal structure is given by 1=(1,,,...)1 = (1, \emptyset, \emptyset, ...).

Relationship to Operads

Symmetric sequences are useful in defining operads (symmetric operads) in symmetric monoidal categories. In particular, an operad in a symmetric monoidal category CC can be defined to be a monoid in the category of symmetric sequences of CC. See, for instance, Definition 2.2.9 of Ching, and see at operad – Definition as monoid.

See also


The last chapter of

  • Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, I. Astérisque, Vol. 314 (2008). Société Mathématique de France. (pdf)

  • Michael Ching, Bar constructions for topological operads and the Goodwillie derivatives of the identity (PhD Thesis), Available Here.

Last revised on July 24, 2016 at 12:16:00. See the history of this page for a list of all contributions to it.