Quasigroupoids and weak Hopf quasigroups

In this paper we prove that the category of finite quasigroupoids is equivalent to the one of pointed cosemisimple weak Hopf quasigroups over a given field K. As a consequence, if K is algebraically closed, we obtain that the categories of finite quasigroupoids and cocommutative cosemisimple weak Hopf quasigroups are equivalent. Moreover the restriction of the previous equivalence to the category of quasigroups (loops with the inverse property) provides a categorical equivalence between the categories of quasigroups and of pointed cosemisimple Hopf quasigroups over K and, as in the weak case, if K is algebraically closed, the category of quasigroups is equivalent to the one of cocommutative cosemisimple Hopf quasigroups. (C) 2020 Elsevier Inc. All rights reserved.