In this paper, we introduce the notion of braiding crossed polymodule and 􀀀,equivariant braided crossed polymodule of polygroups and we give some of their properties. Further, we study the concept of , ber hyperproduct. Our results extend the classical results of crossed squares to crossed polysquares. Moreover, we study crossed polysquare version of homotopy kernels by using the fundamental relations.

The aim of this paper is to introduce the notion of cat1−, groupoids which are the groupoid version of cat1−, groups and to prove the categorical equivalence between crossed modules over groupoids and cat1−, groupoids. In section 4 we introduce the notions of crossed squares over groupoids and of cat2−, groupoids, and then we show their categories are equivalent. These equivalences enable us to obtain more examples of groupoids.

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.

In this paper, we present algebroids and crossed modules of algebroids. We also define pullback crossed module of algebroids.

We introduce an alternative criterion for Morita equivalence over graded tensor categories using equivariant centers and equivariantizations. While Morita equivalence has been extensively studied in the context of fusion categories, primarily through the examination of their centers, recent advancements have broadened its scope to encompass graded tensor categories. This paper presents a novel criterion for characterizing Morita equivalence in graded tensor categories by leveraging equivariant centers and equivariantizations. Notably, the identification of Morita equivalence can be expedited when the Brauer Picard groups are known, offering an efficient approach to establishing the equivalence relationship.To generalize the properties of fusion categories to finite tensor categories, we utilize theconcept of an exact module category, which was introduced by Etingof and Ostrik. Exact module categories offer an intermediary restriction between the semisimplemodule categories of a fusion category and more general cases that may not be semisimpleor finite.

crossed testicular ectopia or transverse testicular ectopia is an extremely rare anomaly in which both testes descend through a single inguinal canal. It is characterized by symptomatic inguinal hernia on one side and cryptorchism on the contralateral side. crossed testicular ectopia is often not diagnosed until surgical exploration. We report a case of crossed testicular ectopia in a 22-month-old boy.

In the definition of a crossed module (T, G,  ), the actions of the group T and G on themselves are given by conjugation. In this paper, we consider these actions to be arbitrary and thus generalize the concept of ordinary crossed module. Therefore, we get the category GCM, of all generalized crossed modules and generalized crossed module morphisms between them, and investigate some of its categorical properties. In particular, we study the relations between epimorphisms and the surjective morphisms, and thus generalize the corresponding results of the category of (ordinary) crossed modules. By generalizing the conjugation action, we can find out what is the superiority of the conjugation to other actions. Also, we can find out that a generalized crossed module with which other actions (other than the conjugation) has the properties similar to a crossed module.

In this paper we will consider the crossed product C (T) × a Z, where T is the unit circle, a (n) = an is a rotation through the angle -2 pnq for n  Î Z, and q is a fixed irrational number. We will apply some results about patial actions to represent this crossed product as a C*-subalgebra of B (L2 (T)). Also, by a different method form the proof of Davidson, we show that this crossed product is isomorphic to the irrational rotation algebra.

In the theory of crossed modules, considering arbitrary self-actions instead of conjugation allows for the extension of the concept of crossed modules and thus the notion of generalized crossed module emerges. In this paperwe give a precise definition for generalized cat1-groups and obtain a functor from the category of generalized cat1-groups to generalized crossed modules. Further, we introduce the notions of coverings and liftings for generalized crossed modules and investigate properties of these structures. Main objective of this study is to obtain an equivalence between the category of coverings and the category of liftings of a given generalized crossed module (𝐴, , 𝐵, , 𝛼, ).

crossed fused ectopic kidneys are a rare renal anomaly. The autopsy prevalence of renal ectopia is more than clinical prevalence. Most presenting symptoms of crossed ectopic kidneys are nonspecific and most cases remain asymptomatic through their life and are diagnosed incidentally. The presented case was a 5 years old girl who suffered from episodic left flank pain that imaging findings on sonography and also technetium-99m dimercaptosuccinic acid (DMSA) scintigraphy indicated crossed fused ectopic kidneys.