Pre-torsors and equivalences

@inproceedings{Bhm2006PretorsorsAE,
  title={Pre-torsors and equivalences},
  author={Gabriella B{\`o}hm and Tomasz Brzezinski},
  year={2006}
}
Properties of (most general) non-commutative torsors or A–B torsors are analysed. Starting with pretorsors it is shown that they are equivalent to a certain class of Galois extensions of algebras by corings. It is shown that a class of faithfully flat pre-torsors induces equivalences between categories of comodules of associated corings. It is then proven that A–B torsors correspond to monoidal functors (and, under some additional conditions, equivalences) between categories of comodules of… 
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References

SHOWING 1-10 OF 52 REFERENCES
Pseudo-Galois Extensions and Hopf Algebroids
A pseudo-Galois extension is shown to be a depth two extension. Studying its left bialgebroid, we construct an enveloping Hopf algebroid for the semi-direct product of groups, or more generally
On hopf algebras and rigid monoidal categories
LetC be a neutral Tannakian category over a fieldk. By a theorem of Saavedra Rivano there exists a commutative Hopf algebraA overk such thatC is equivalent to the category of finite dimensional
Quantum torsors with fewer axioms
We give a definition of a noncommutative torsor by a subset of the axioms previously given by Grunspan. We show that noncommutative torsors are an equivalent description of Hopf-Galois objects
Fibre functors of finite dimensional comodules
Let A be a commutative Hopf algebra over a field k; the k-valued fibre functors on the category of finite dimensional A-comodules correspond to Spec(A)-torsors over k as was shown by Saavedra Rivano
Coalgebra extensions and algebra coextensions of galois type
The notion of a coalgebra-Galois extension is defined as a natural generalisation of a Hopf-Galois extension. It is shown that any coalgebra-Galois extension induces a unique entwining map ψ
Some Bialgebroids Constructed by Kadison and Connes–Moscovici are Isomorphic
TLDR
It is proved that a certain bialgebroid introduced recently by Kadison is isomorphic to a bial Geometric Algebra introduced earlier by Connes and Moscovici.
Galois extensions as functors of comodules
Let A be a finite Hopf algebra over a commutative ring k. We show a one-to-one correspondence between the A-Galois extensions of k and certain functors from the category of A-comodules to the
Cleft Extensions of Hopf Algebroids
TLDR
It is shown that an extension with a Hopf algebroid is cleft if and only if it is ℋR-Galois and has a normal basis property relative to the base ring L ofℋL.
Hopf algebroids and quantum groupoids
We introduce the notion of Hopf algebroids, in which neither the total algebras nor the base algebras are required to be commutative. We give a class of Hopf algebroids associated to module algebras
Principal homogeneous spaces for arbitrary Hopf algebras
LetH be a Hopf algebra over a field with bijective antipode,A a rightH-comodule algebra,B the subalgebra ofH-coinvariant elements and can:A ⊗BA →A ⊗H the canonical map. ThenA is a faithfully flat (as
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