The Galois correspondence theorem for groupoid actions

Descent and Galois theory for Hopf categories

“…Indeed, applying can y xy ⊗H yx to (24), we obtain (25). Applying ρ yy ⊗((S xy ⊗ H xy ) • ∆ xy ) to (17), we obtain that (2) . Now we multiply the second and third tensor factor, and obtain that (2) .…”

“…Recall that i x : B x → A xx , so that i x • a * j ∈ Bx Hom(A xy , A xx ). We have to show that (17)(18) are satisfied. (18) follows from the fact thatδ x xy is a right inverse of δ x xy : for all f ∈ Bx Hom(A xy , A xx ) and a ∈ A xy , we have [1] ), hence i a [0] l i (a [1] ) ⊗ Bx r i (a [1] )…”

Isomorphism Theorems for Groupoids and Some Applications

“…Also, ring theoretic and cohomological results of global and partial actions of groupoids on algebras are obtained in [10][11][12][13][14][15][16]. Galois theoretic results for groupoid actions are obtained in [12,[17][18][19]. Finally, the globalization problem for partial groupoid actions has been considered in [7,20,21].…”

“…In [19], Paques and Tamusiunas give some structural definitions in the context of groupoid such as abelian groupoid, subgroupoid, and normal subgroupoid and showed necessary and sufficient conditions for a subgroupoid to be normal. Furthermore, they built quotient groupoids.…”

“…Remark 4. A concept of abelian groupoid was presented in ( [19], p. 111) as follows: a groupoid G is abelian if d(g) � r(g) for each g ∈ G; and gh � hg, for all g, h ∈ G with d(g) � r(h).…”