  A new quantization of groupoids under the name of \times-Hopf coalgebras is
introduced. We develop a Hopf cyclic theory with coefficients in
stable-anti-Yetter-Drinfeld modules for \times-Hopf coalgebras. We use
\times-Hopf coalgebras to study coextensions of coalgebras. Finally,
equivariant \times-Hopf coalgebra Galois coextensions are defined and applied
as functors between categories of stable anti-Yetter-Drinfeld modules over
\times-Hopf coalgebras involved in the coextension.

Quantum Algebra