IAPS of groups

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This article gives a basic definition in the following area: APS theory
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This article defines the notion of group object in the category of IAPSs|View other types of group objects

Definition

An IAPS of groups is an IAPS over the category of groups. More specifically an IAPS $(G,\mathrm{\Phi })$ is the following data:

• For each natural number $n$, a group denoted $G_n$
• For each ordered pair $(m,n)$ of natural numbers, an injective 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.

Note that if we remove the condition of injectivity, we get an APS of groups.

Constructions

Sub-IAPS

Further information: sub-IAPS of groups

Let $(G,\mathrm{\Phi })$ be an IAPS of groups. A sub-IAPS $H$ associated to every $n$ a subgroup $H_n$ of $G_n$ such that the image of $H_m\times H_n$ under ${\mathrm{\Phi }}_{m,n}$ lies inside $H_{m+n}$. Note that this is the same as a sub-APS of groups because the injectivity condition comes for free.

Quotient IAPS

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