Normal sub-APS

Definition
Let $$(G,\Phi)$$ be an APS of groups and $$H$$ be a sub-APS of $$G$$. We say that $$H$$ is normal in $$G$$ if the following equivalent conditions hold:


 * For every $$n$$, $$H_n$$ is a normal subgroup of $$G_n$$
 * There exists an APS of groups $$(K,\Psi)$$ and an APS homomorphism from $$(G,\Phi)$$ to $$(K,\Psi)$$ such that the kernel of the map at each $$n$$ is $$H_n$$

Stronger properties

 * Saturated normal sub-APS