%0 Journal Article %T Cat$^1$-polygroups and pullback cat$^1$-polygroups %J Bulletin of the Iranian Mathematical Society %I Iranian Mathematical Society (IMS) %Z 1017-060X %A Davvaz, Bijan %A Alp, Murat %D 2014 %\ 06/01/2014 %V 40 %N 3 %P 721-735 %! Cat$^1$-polygroups and pullback cat$^1$-polygroups %K polygroup %K crossed polymodule %K cat$^1$-group %K cat$^1$-polygroup %K pullback cat$^1$-polygroup %R %X 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. %U http://bims.iranjournals.ir/article_527_6286c7397d9d3edb57e3007f8052de19.pdf