process

362 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganLet Algd R denote the 2-category of R-algebroids (full 2-subcategory of the 2-categoryof R-linear categories).Suppose that A is an R-algebra. The R-linear category with one object and morphismsA is an R-algebroid denoted A C .Suppose that C is an R-algebroid and L is an object of C. Let A D End C .L/. Thefunctor A C ! C which sends the unique object of A C to L is an equivalence.Let Alg 2 Rdenote the 2-category with• objects: R-algebras,• 1-morphisms: homomorphism of R-algebras,• 2-morphisms ! , where ; W A ! B are two 1-morphisms: elementsb 2 B such that .a/ b D b .a/ for all a 2 A.It is clear that the 1- and the 2- morphisms in Alg 2 Ras defined above induce 1- and2-morphisms of the corresponding algebroids under the assignment A 7! A C . Thestructure of a 2-category on Alg 2 R(i.e. composition of 1- and 2-morphisms) is determinedby the requirement that the assignment A 7! A C extends to an embedding./ C W Alg 2 R ! Algd R .Suppose that R ! S is a morphism of commutative k-algebras. The assignmentA ! A ˝R S extends to a functor ./ ˝R S W Alg 2 R ! Alg2 S .4.1.2 Algebroid stacks. We refer the reader to [1] and [20] for basic definitions. Wewill use the notion of fibered category interchangeably with that of a pseudo-functor.A prestack C on a space X is a category fibered over the category of open subsetsof X, equivalently, a pseudo-functor U 7! C.U /, satisfying the following additionalrequirement. For an open subset U of X and two objects A; B 2 C.U / we have thepresheaf Hom C .A; B/ on U defined by U V 7! Hom C.V / .Aj V ;Bj B /. The fiberedcategory C is a prestack if for any U , A; B 2 C.U /, the presheaf Hom C .A; B/ is asheaf. A prestack is a stack if, in addition, it satisfies the condition of effective descentfor objects. For a prestack C we denote the associated stack by zC.Definition 4.2. A stack in R-linear categories C on X is an R-algebroid stack if it islocally nonempty and locally connected, i.e. satisfies1. any point x 2 X has a neighborhood U such that C.U / is nonempty;2. for any U X, x 2 U , A; B 2 C.U / there exits a neighborhood V U of xand an isomorphism Aj V Š Bj V .Remark 4.3. Equivalently, the stack associated to the substack of isomorphisms iC isa gerbe.Example 4.4. Suppose that A is a sheaf of R-algebras on X. The assignment X U 7! A.U / C extends in an obvious way to a prestack in R-algebroids denoted A C .