Imagine a grounding functor U: \({\mathfrak A}\to {\mathfrak B}\) (”U forgets some of the structure extant in \({\mathfrak A}'')\) and contemplate objects \(\Phi =(X,F,(A_ x)_{x\in X}),\) where X is a topological space, F: O(X)\({}^{op}\to {\mathfrak B}\) a sheaf in \({\mathfrak B}\), and \(A_ x\) a family of objects in \({\mathfrak A}\) such that each stalk \(F_ x\) of the sheaf may be identified with \(UA_ x:\) ”a sheaf in \({\mathfrak B}\) with stalks \(in\quad {\mathfrak A} !''.\) The guiding example is the affine scheme of a commutative ring for which \({\mathfrak A}\) is the category of local rings and \({\mathfrak B}\) the category of rings (all commutative and with identity). If one knows how to define morphisms between sheaves then one knows how to make the objects \(\Phi\) into a category SH\({\mathfrak AB}\). The global section functor \(\Gamma\) associates with a \(\Phi\) a \({\mathfrak B}\)- object \(\Gamma_{\Phi}(X)\) and a family of \({\mathfrak B}\)-morphisms \(ev_ x: \Gamma_{\Phi}(X)\to UA_ x,\quad x\in X.\)
Meanwhile, back at the level of pure category theory, the functor U: \({\mathfrak A}\to {\mathfrak B}\) is said to have a multiadjoint on the left if for each object B of \({\mathfrak B}\) there is a family \(\eta_ x: B\to UA_ x,\quad A_ x\in ob {\mathfrak A},\quad x\in X\) of \({\mathfrak B}\)-morphisms such that for every \({\mathfrak B}\)-morphism f: \(B\to UA\) there is a unique \(x\in X\) and a unique \({\mathfrak A}\)-morphism \(f': A_ x\to A\) so that \((Uf') \eta_ x=f.\) Under these circumstances, the set X is called the (U-)spectrum of B, written \(Spec_ U B\). The author’s objective: Given a grounding functor U: \({\mathfrak A}\to {\mathfrak B}\) allowing a multiadjoint on the left, construct a topology on \(Spec_ U B\) and a sheaf \(F: O(Spec_ U B)^{op}\to B\) with stalks \(UA_ x\) (coming from the adjoint!) such that \(LB=(Spec_ U B,F,(A_ x)_{x\in Spec_ U B})\) is a universal SH\({\mathfrak AB}\)-object, i.e., that L is a left adjoint for the global section functor. The author finds sufficient conditions which allow him to prove two theorems on the existence of such a construction, and he tests the generality of his results by listing 38 concrete situations in which it applies.
It seems to be a real challenge to find new and more general principles of a general sectional representation theory. This one is novel, and it works in situations where sectional representation is anyhow known to work. It does not cover all the situations in which sectional representation is possible, and like all general theories, it does not provide a general answer to the question when the Gelfand morphism \(B\to \Gamma_{LB}(Spec_ U B)\) is an isomorphism; in the concrete situations the quality of a sectional representation depends on an answer to this question. On the other hand, the numerous examples in which an object in a category is represented by its ”cofinite” quotients are no very convincing application of a formidable categorical and sheaf theoretical machinery as long as simple projective limits fill the bill.