In the framework of the EU/ESA co-funded Copernicus program, ESA is undertaking the development of a series of five Sentinel missions with the objective to provide routinely Earth observation data for the implementation of operational Copernicus and national services. The Copernicus services comprise mapping and forecasting activities for Land, Marine, Atmosphere, Emergency, Security, and Clima...

in this paper‎, ‎we give the notions of crossed polymodule and cat$^1$-polygroup‎ ‎as a generalization of loday's definition‎. ‎then‎, ‎we define the pullback cat$^1$-polygroup and we obtain some results in this respect‎. ‎specially‎, ‎we prove that by a pullback cat$^1$-polygroup we can obtain a cat$^1$-group‎.