A factorization system inter alia pertaining to descent

Ross Street – 14 August 2002

The talk revisited the last few pages of my paper "Characterization of bicategories of stacks", Lecture Notes in Math. 962 (1982) 282-291 dealing with stacks and torsors. I will restrict myself here with describing the factorization system alluded to in the title of the talk. Let E be a category with some distinguished morphisms called "covers" satisfying a few simple axioms (like regular epimorphisms in a regular category). Let F = Hom (Eop , Cat) be the 2-category of "categories varying over E" (that is, the objects are pseudofunctors from Eop to Cat). A morphism j : X --> Y in F is called ccff (meaning "cover cartesian and fully faithful") when its components j_U : XU --> YU are all fully faithful and the squares expressing pseudonaturality, that involve a morphism of E which is a cover V --> U, are pseudopullbacks. A morphism s : A --> B in F is called lso (meaning "locally surjective on objects") when, for all objects b of each BU, there exists a cover e : V --> U and a in AV such that (s_V)a is isomorphic to (Be)b. These two classes of morphism determine a factorization system on the bicategory F. Each morphism f : A --> X in F factors pseudofunctorially as A --> Z --> X where ZU consists of those x in XU that are "locally isomorphic to an object in the image of f_U ". In particular, for a category A in E, this factorization applied to the "yoneda embedding of A" yields the stack Tors(A) of torsors over A.