in this paper, we present algebroids and crossed modules of algebroids. we also define pullbackcrossed module of algebroids.

‎In this paper‎, ‎we prove that the category of crossed polymodules (i.e‎. ‎crossed modules of polygroups) and their morphisms is finitely complete‎. ‎We‎, ‎therefore‎, ‎generalize the group theoretical case of this completeness property of crossed modules‎.

1 Crossed Modules and Cat1-Groups 4 1.1 Pre-crossed and Crossed Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Examples of Crossed Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Properties of Crossed Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Sub-crossed modules . . . . . . . . . . . . . . . . . . . . . ....