This survey article shows how the notion of “change of base”, used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids.

The concept of Smarandache isotopy is introduced and its study is explored for Smarandache: groupoids, quasigroups and loops just like the study of isotopy theory was carried out for groupoids, quasigroups and loops. The exploration includes: Smarandache; isotopy and isomorphy classes, Smarandache f, g principal isotopes and G-Smarandache loops.

The Grothendieck rings of finite dimensional representations of the basic classical Lie superalgebras are explicitly described in terms of the corresponding generalised root systems. We show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids, which we call Weyl groupoids.

A bstract We review the properties of orbifold operations on two-dimensional quantum field theories, either bosonic or fermionic, and describe “Orbifold groupoids” which control composition operations. Three-dimensional TQFT’s Dijkgraaf-Witten type will play an important role in analysis. briefly discuss extension to generalized symmetries applications constrain RG flows.

In this paper, we study a family of C*-subalgebras defined by fixed points generalized gauge actions Cuntz–Krieger algebra introducing étale groupoids whose associated C*-algebras are these C*-subalgebras. We show that topological conjugacy classes one-sided Markov shifts characterized in terms the isomorphism groupoids.