Let L be a Lie CROSSED MODULE and Actpi(L) and Actz(L) be the pointwise inner actor and center actor of L, respectively. We will give a necessary and sufficient condition under which Actpi(L) and Actz(L) are equal.

Using free simplicial algebras with given CW−basis, it is shown how to construct a free or totally free quasi 3-CROSSED MODULE on suitable construction data. Quasi 3-CROSSED complexes are introduced and similar freeness results are given for these are discussed.

In this paper, we present algebroids and CROSSED MODULEs of algebroids. We also define pullback CROSSED MODULE of algebroids.

In this work we deal with actions of vector groupoid which is a new concept in the literature. After we give the definition of the action of a vector groupoid on a vector space, we obtain some results related to actions of vector groupoids. We also apply some characterizations of the category and groupoid theory to vector groupoids. As the second part of the work, we define the notion of a CROSSED MODULE over a vector groupoid. Finally, we show that the category Vg of the vector groupoids is equivalent to the category CModVg of the CROSSED MODULEs over a vector groupoid.

In this paper, we presented algebroids and CROSSED MODULEs of algebroids. We also presented pullback CROSSED MODULE of algebroids and we then define pushout CROSSED MODULEs of algebroids.

The theory of polygroups is a natural extension of group theory, where the composition of two elements results in a set rather than an element. This concept has found applications in diverse fields such as geometry, lattices, combinatorics, and color schemes. Addition-ally, the study of CROSSED MODULEs and their applications has played a crucial role in category theory, homology and cohomology of groups, homotopy theory, algebra, k-theory, and more. This paper presents the definition of a polyfunctor and transformation for polygroups, as well as the introduction of the concept of symmetric CROSSED MODULE to sym-metric CROSSED polyMODULEs. Our findings extend the classical results of CROSSED MODULEs to CROSSED polyMODULEs of polygroups.