nLab symmetric sequence

Idea

A symmetric sequence is a sequence of objects where the $n$th object has an action of the $n$th symmetric group.

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

Definition

Definition

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

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

3. A $\Phi$-symmetric sequence in $C$ is a sequence of $\Phi_n$-representations for $n \ge 0$:

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

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

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

Definition

A $\Sigma$-symmetric sequence in a symmetric monoidal category $C$ is a functor from $FinSet$, the category of finite sets and bijections, to $C$.

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

Symmetric monoidal structure

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

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

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

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

   called the induced representation functor.

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

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

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

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 $C$ can be defined to be a monoid in the category of symmetric sequences of $C$. See, for instance, Definition 2.2.9 of Ching, and see at operad – Definition as monoid.

See also

• spectra, symmetric spectra

• colored symmetric sequence

References

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.