APS of groups

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Definition

An APS of groups is an APS (short for additively parametrized sequence) over the category of groups. More specifically, an APS $(G,\mathrm{\Phi })$ of groups is the following data:

• For each natural number $n$, a group, denoted $G_n$.
• For each ordered pair $(m,n)$ of natural numbers, a homomorphism ${\mathrm{\Phi }}_{m,n}:G_m\times G_n\to G_{m+n}$.

Satisfying the following compatibility conditions:

For $g,h,k$ in $G_m,G_n,G_p$ respectively:

${\mathrm{\Phi }}_{m+n,p}({\mathrm{\Phi }}_{m,n}(g,h),k)={\mathrm{\Phi }}_{m,n+p}(g,{\mathrm{\Phi }}_{n,p}(h,k))$.

The above condition is termed an associativity condition.

We may assume $G_0$ as the trivial group and define ${\mathrm{\Phi }}_{m,0}$ and ${\mathrm{\Phi }}_{0,n}$ as trivial paddings.

An APS can be thought of loosely as a $\mathbb{N}$-graded family of groups; however, the jargon "graded group" has a somewhat different meaning in the world of abelian groups and modules, hence this term.

Relation with other literature concepts

• Imposing both additive and multiplicative structure (which is much richer than the purely additive structure described here) yields what is called a bipermutative category, a concept developed by May and Elmendorf–Mandell.
• An APS of groups is equivalently a lax monoidal functor from the discrete monoidal category $(\mathbb{N},+,0)$ to $(Grp,X,1)$, i.e. a monoid object in $\mathbb{N}$-graded groups under Day convolution.
• It is not an operad, though it is an algebra over the associative operad in graded objects; the operad formalism carries structure (multi-input composition, symmetric actions) not used here.
• An inner-permutative IAPS is close to the braided monoidal groupoid of Randal-Williams–Wahl, Homological stability for automorphism groups (Adv. Math. 318, 2017), where the symmetric groups and general linear groups are the motivating examples. Applying the representation-ring functor to an IAPS yields the PSH algebras of Zelevinsky, Representations of Finite Classical Groups: A Hopf Algebra Approach (LNM 869, 1981). The terminology here is retained because the derived notions (saturated and contrasaturated, i-simplicity, unique class factorization) have no located standard equivalents.

Terminology

Members and elements

For an APS $(G,\mathrm{\Phi })$ the member $G_n$ is termed the $n^{th}$ member of the APS. A member of the APS is an object that is the $n^{th}$ member for some $n$.

An element of the APS is an element of some member of the APS.

The home of an element of the APS is the member in which it lies. The index of a member is the $n$ for which it is the $n^{th}$ member.

Block concatenation map

The maps ${\mathrm{\Phi }}_{m,n}$ are termed block concatenation maps.

Ground member

The ground member of an APS is its first member.

Other notions

Homomorphism of APSes

Further information: APS homomorphism

Given APSes $(G,\mathrm{\Phi })$ and $(H,\mathrm{\Psi })$, a homomorphism $h:G$ → $H$ associates, to each natural number $n$, a map $h_n:G_n$ → $H_n$, such that:

${\mathrm{\Psi }}_{m,n}(h_m(a),h_n(b))=h_{m+n}({\mathrm{\Phi }}_{m,n}(a,b))$

Sub-APS notion

Further information: sub-APS

Given an APS $(G,\mathrm{\Phi })$, a sub-APS $H$ associates, to each $n$, a subgroup $H_n$ of $G_n$, such that the image of $H_{m}X{H}_n$ under ${\mathrm{\Phi }}_{m,n}$ lies inside $H_{m+n}$.

When the APS of groups is injective, any sub-APS is also injective.

Quotient APS notion

Further information: quotient APS

A quotient APS is the image of an APS in a homomorphism that is surjective at each member.

Normal sub-APSes, kernels and images

Given a homomorphism of APSes of groups, the kernels of the individual homomorphisms for a sub-APS of the domain APS, and the images of the individual homomorphism form a sub-APS of the range APS. The image is thus a quotient APS.

Further, we have the following result: a sub-APS of an APS of groups occurs as the kernel of an APS homomorphism if and only if every member of it is a normal subgroup of the corresponding member of the whole APS. A sub-APS satisfying either of these equivalent conditions is termed a normal sub-APS.

This parallels the group theory result that a subgroup of a group occurs as the kernel of a group homomorphism if and only if it is normal.

Properties

Injectivity

An APS of groups is termed injective, or an IAPS of groups, if every block concatenation map is injective. For an IAPS of groups, we usually also assume the condition of refinability.

Commutativity

Very few APSes of groups are commutative. Note that a commutative APS cannot also be injective.

Padding-injectivity

Most APSes of groups that we encounter satisfy the condition of being padding-injective.