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.
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 $C$.
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.
A $\Phi$-symmetric sequence in $C$ is a sequence of $\Phi_n$-representations for $n \ge 0$:
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:
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.”
Let $\alpha : H \to G$ be a homomorphism of groups and consider the restriction of scalars functor
which is defined in the obvious way. It admits a left adjoint
called the induced representation functor.
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
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 $1 = (1, \emptyset, \emptyset, ...)$.
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.
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 16:16:00. See the history of this page for a list of all contributions to it.