Bisections and cocycles on Hopf algebroids

We introduce and study the group \\mathcal{B}(\\mathcal{L}) of bisections of a Hopf algebroid \\mathcal{L} and show that they form a group crossed module or 2 -group with the group \\mathrm{Aut}(\\mathcal{L}) of automorphisms. Moreover, the group of vertical bisections turns out to be part of a certain non-Abelian cohomology \\mathcal{H}^{2}(\\mathcal{L},B) governing cotwisting of a Hopf algebroid with base B . For the Ehresmann–Schauenburg Hopf algebroid \\mathcal{L}(P,H) of a quantum principal bundle or Hopf–Galois extension, \\mathcal{B}(\\mathcal{L}(P,H)) reduces to the group \\operatorname{Aut}_{H}(P) of bundle automorphisms and vertical bisections to the group of ‘gauge transformations’ of the bundle. The general \\mathcal{H}^{2}(\\mathcal{L}(P,H),B) reduces to a known non-Abelian cohomology in the case where P is a trivial principal bundle or cleft extension. Parallel characterisations are obtained for the bisections and non-Abelian cohomology of the action Hopf algebroid B\\# H^{\\mathrm{op}} associated with a braided-commutative algebra B in the category of Drinfeld–Yetter modules over a Hopf algebra H . Examples include the Heisenberg double or Weyl Hopf algebroid of a Hopf algebra and a canonical action Hopf algebroid \\underline{H}\\# H^{\\mathrm{op}} when H is coquasitriangular and \\underline{H} is its transmutation.