From Grothendieck’s foundation, the setting for a descent theory is given by a fibration \(\Phi:{\mathcal D}\to{\mathcal C}\) and structures over an object \(C\) are given by the fibre \(\Phi^{-1}(C)\). In the present paper the authors consider only the basic fibration of \(C\), i.e., the codomain functor \({\mathcal C}^2\to{\mathcal C}\), the fibre of \({\mathcal C}\) being the comma category \(({\mathcal C}\downarrow C)\) and a structure over \({\mathcal C}\) is simply a morphism in \({\mathcal C}\) with codomain \(C\). Then effective descent morphisms \(p:B\to C\) in \({\mathcal C}\) are defined and studied which will facilitate an algebraic description of \(({\mathcal C} \downarrow C)\) by means of \(({\mathcal C}\downarrow B)\). For a Barr exact category, these are the regular epimorphisms. One has then a complete presentation of (basic) descent theory including sheaf-theoretic characterization, links with other categorical constructions (monads, Grothendieck’s theory, internal category actions,…) and the case of \({\mathcal C}={\mathcal C}at\). The way is open towards applications in homological algebra, Galois theory and algebraic topolopy.
For the entire collection see [Zbl 1034.18001].