Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

2<strong>1.</strong> <strong>Introduction</strong>The starting point of this work was the notion of a torsor which comes from principlebundles in classical geometry. A torsor is a principal homogeneous space overa group, i.e. a G-set X where G is a group acting freely and transitively over X. Anidea due to Baer which goes back to the 1920’s allows to reformulate the definitionof a torsor without specifying the group G: a torsor is a set, sometimes called herd,X together with a structure X ×X ×X → X satisfying some parallelogram relations(see [Ba, p. 202] or [Pr, p. 170]).A noncommutative analogue is the notion of a Hopf-Galois object as introducedby Kreimer and Takeuchi [KT]. Let H be a Hopf algebra, flat over the basering k, a (right) H-Galois object A is a (right) H-comodule algebra such that theGalois map β : A ⊗ A → A ⊗ H given by β (x ⊗ y) = xy (0) ⊗ y (1) is bijective(where δ : A → A ⊗ H : x ↦→ x (0) ⊗ x (1) is the H-comodule structure of A) andA co(H) = {x ∈ A | δ (x) = x ⊗ 1 H } = k.A similar concept for the noncommutative case was introduced by Grunspan in [G]as the notion of a quantum torsor. Together with the definition, Grunspan givesthe proof that every quantum torsor gives rise to two Hopf algebras over which itis a bi-Galois extension of the base field. Conversely, Schauenburg in [Sch4] provesthat every Hopf-Galois extension of the base field is a quantum torsor in the senseof Grunspan.The axioms defining a quantum torsor were simplified allowing to prove anywaya correspondence between faithfully flat torsors and Hopf-(bi)Galois object (see[Sch1]). Moreover, Schauenburg in [Sch4] could prove that the two Hopf algebrascoming from a torsor are Morita-Takeuchi equivalent, i.e. their categories of comodulesare equivalent. Another equivalence between module categories has beenstudied in [BMV] and it is related with Morita contexts defined in the pure categoricalsetting. This gave the hint to investigate a special class of herds at this level ofgenerality.The simplified version of the torsor axioms admits a generalization to arbitraryGalois extensions (not only of the base ring or field) and gave rise to different <strong>results</strong>which we try to summarize here. Hopf Galois extensions of an arbitrary algebra Bby introducing the notion of a B-torsor in [Sch1], Galois extensions by bialgebroidsby means of A-B-torsors in [Ho, Chapter 5] and [BB], Galois extensions by coringsusing the notion of a pretorsor in [BB] and Galois comodules of corings arising fromentwining structures using the notion of a bimodule herd in [BV]. Generalizing thenotion of pretorsor given in [BB], pretorsors over two adjunctions are introduced in[BM, Section 4]. Such categorical setting is the one we choose for this work tryingto develop the notion of pretorsor and herd at this pure general level.The first aim of this thesis is to give a unified and self-contained treatment of anumber of known <strong>results</strong> related to the theory of herds. This gives us the technicaltools to deal with the second aim of our work which is to obtain some new <strong>results</strong>about herds and coherds in the pure categorical setting. A herd at this level is apretorsor with respect to a formal dual structure M = (A, B, P, Q, σ A , σ B ). This isgiven by two monads A and B over two different categories, two bimodule functors Pand Q with respect to the monads and functorial morphisms σ A and σ B satisfying