IAPS of groups

Definition
An IAPS of groups is an IAPS over the category of groups. More specifically an IAPS $$(G,\Phi)$$ 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 $$\Phi_{m,n}:G_m \times G_n \to G_{m+n}$$

Satisfying the following compatibility conditions:

For $$g, h, k$$ in $$G_m, G_n, G_p$$ respectively: $$\Phi_{m+n,p} (\Phi_{m,n}(g,h),k) = \Phi_{m,n+p} (g, \Phi_{n,p}(h,k))$$.

The above condition is termed an associativity condition.

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

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

Sub-IAPS
Let $$(G,\Phi)$$ 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 $$\Phi_{m,n}$$ lies inside $$H_{m+n}$$. Note that this is the same as a sub-APS of groups because the injectivity condition comes for free.