Localization of universal problems. Local colimits
Let \({\mathcal A}\) be a category and \({\mathcal B}\), \({\mathcal B}'\) two full subcategories of \({\mathcal A}\). An \({\mathcal A}\)-cylinder from \({\mathcal B}\) to \({\mathcal B}'\) is a couple \((F,f)\) of a map \(F:\text{Obj}({\mathcal B})\to \text{Obj} ({\mathcal B}')\) and a map \(f:\text{Obj}({\mathcal B})\to \text{Mor} ({\mathcal B}')\) such that, for any \(b\in\text{Obj}({\mathcal B})\), we have \(f(b):b\to F(b)\), and, for any \(x:b\to c\) in \(\text{Mor} ({\mathcal B})\), there exists some morphism \(x':F(b) \to F(c)\) such that \(x'f(b)= f(c)x\). The category \({\mathcal B}\) is called a root of \({\mathcal A}\) if there exists at least one \({\mathcal A}\)-cylinder from \({\mathcal A}\) to \({\mathcal B}\), and if any \({\mathcal A}\)-cylinder from \({\mathcal B}\) to \({\mathcal B}\) is invertible in the category \(\text{cyl} {\mathcal B}\) of cylinders of \({\mathcal B}\). Let \(U:{\mathcal C}'\to {\mathcal C}\) be a functor. A \(U\)-universal root generated by an object \(c\in\text{Obj}({\mathcal C})\) is a root of the comma category \(c/U\). Whenever it exists, it is unique, up to isomorphism. Whenever it exists for any \(c\in \text{Obj} ({\mathcal C})\), it defines an adjoint-root functor, taking its values in the category of pro-objects of \({\mathcal C}'\). Local colimits are obtained if \(U\) is the insertion from a category into its category of ind-objects; they generalize Diers' multicolimits. Applications to posets and Galois theory are given.
scientific article; zbMATH DE number 1783485