APS-on-APS action

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This term is related to: APS theory
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Let (G,\Phi) be an APS of groups and (T,\Psi) an APS of sets. An action of G on T is defined as the data of an action of G_n on T_n for every n, such that for any natural numbers m,n and any elements g \in G_m, h \in G_n, s \in T_m, t \in T_n, we have:

\Phi_{m,n}(g,h) \cdot \Psi_{m,n}(s,t) = \Psi_{m,n}(g \cdot s, h \cdot t)