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In this paper we introduce the notion of weak non-asssociative Doi-Hopf module and give the Fundamental Theorem of Hopf modules in this setting. Also we prove that there exists a categorical equivalence that admits as particular instances the ones constructed in the literature for Hopf algebras, weak Hopf algebras, Hopf quasigroups, and weak Hopf quasigroups.

In this paper we introduce the theory of multiplication alteration by two-cocycles for nonassociative structures like nonassociative bimonoids with left (right) division. Also we explore the connections between Yetter-Drinfeld modules for Hopf quasigroups, projections of Hopf quasigroups, skew pairings, and quasitriangular structures, obtaining the nonassociative version of the main results proved by Doi and Takeuchi for Hopf algebras.

Let H be a cocommutative faithfully flat Hopf quasigroup in a strict symmetric monoidal category with equalizers. In this paper we introduce the notion of (strong) Galois H-object and we prove that the set of isomorphism classes of (strong) Galois H-objects is a (group) monoid which coincides, in the Hopf algebra setting, with the Galois group of H-Galois objects introduced by Chase and Sweedler.

In this paper we show that weak Hopf (co)quasigroups can be characterized by a Galois-type condition. Taking into account that this notion generalizes the ones of Hopf (co)quasigroup and weak Hopf algebra, we obtain as a consequence the first fundamental theorem for Hopf (co)quasigroups and a characterization of weak Hopf algebras in terms of bijectivity of a Galois-type morphism (also called fusion morphism)., Comment: arXiv admin note: text overlap with arXiv:1410.2180

In this paper we show how iterate weak crossed products with common monoid. More concretely, if $(A\otimes V, \mu_{A\otimes V})$ and $(A\otimes W, \mu_{A\otimes W})$ are weak crossed products, we find sufficient conditions to obtain a new weak crossed product $(A\otimes V\otimes W, \mu_{A\otimes V\otimes W})$

In this paper we introduce the notions of cleft and Galois (with normal basis) extension associated to a weak Hopf quasigroup. We show that, under suitable conditions, both notions are equivalent. As a particular instance we recover the classical results for (weak) Hopf algebras. Moreover, taking into account that weak Hopf quasigroups generalize the notion of Hopf quasigroup, we obtain the definitions of cleft and Galois (with normal basis) extension associated to a Hopf quasigroup and we get the equivalence betwen these extensions in this setting., Comment: arXiv admin note: text overlap with arXiv:1410.2180

In this paper we introduce the notion of weak Hopf quasigroup as a generalization of weak Hopf algebras and Hopf quasigroups. We obtain its main properties and we prove the fundamental theorem of Hopf modules for these algebraic structures.

In this paper we present the general theory of cleft extensions for a cocommutative weak Hopf algebra $H$. For a weak left $H$-module algebra we obtain a bijective correspondence between the isomorphisms classes of $H$-cleft extensions $A_{H}\hookrightarrow A$, where $A_{H}$ is the subalgebra of coinvariants, and the equivalence classes of crossed systems for $H$ over $A_H$. Finally, we establish a bijection between the set of equivalence classes of crossed systems with a fixed weak $H$-module algebra structure and the second cohomology group $H_{\varphi_{Z(A_H)}}^2(H, Z(A_H))$, where $Z(A_H)$ is the center of $A_H$.

In this paper we present the Sweedler cohomology for a cocommutative weak Hopf algebra H. We show that the second cohomology group classifies completely the weak crossed products, having a common preunit, of H with a commutative left H-module algebra A.

In this paper we introduce the notion of weak operator and the theory of Yetter-Drinfeld modules over a weak braided Hopf algebra with invertible antipode in a strict monoidal category. We prove that the class of such objects constitutes a non strict monoidal category. It is also shown that this category is not trivial, that is to say that it admits objects generated by the adjoint action (coaction) associated to the weak braided Hopf algebra.

In [J.M. Fern\'andez Vilaboa, R. Gonz\'alez Rodr\'iguez and A.B. Rodr\'iguez Raposo: Preunits and weak crossed products. J. of Pure Appl. Algebra 213, 2244-2261 (2009)] the notion of a weak crossed product of an algebra by an object, both living in a monoidal category was presented. Unified crossed products defined in [A. Agore, G. Militaru: Extending structures II: The quantum version. arXiv:1011.2174v3 (2011)] and partial crossed products defined in [M. Muniz S. Alves, E. Batista, M. Dokuchaev, A. Paques: Twisted partial actions of Hopf algebras. preprint (2011)] are crossed product structures defined for a Hopf algebra and another object. In this paper we prove that unified crossed products and partial crossed products are particular instances of weak crossed products.

We present the universal theory of weak crossed biproducts, and we prove that every weak projection of weak bialgebras induces an example of this crossed structure. As an example, we give the construction of a weak projection of a weak bialgebra associated to a groupoid that admits an exact factorization.