process

366 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganSuppose that .i/ W .U; A/ ! .U; B/ and .i/ W .U; B/ ! .U; C/, i D 1; 2,are 1-morphisms and b W .1/ ! .2/ and c W .1/ ! .2/ are 2-morphisms. Thesection c .1/ .b/ 2 .N 0 UI C/ defines a 2-morphism, denoted c ˝ b, the horizontalcomposition of b and c.We leave it to the reader to check that with the compositions defined above descentdata, 1-morphisms and 2-morphisms form a 2-category, denoted Desc R .U/.4.2.6 Fibered category of descent data. Suppose that W V ! U is a morphism ofcovers and .U; A/ is a descent datum. Let A 01 D .N 1/ A 01 , A 012 D .N 2/ .A 012 /.Then, .V; A / is a descent datum. The assignment .U; A/ 7! .V; A / extends to afunctor, denoted W Desc R .U/ ! Desc R .V/.The assignment Cov.X/ op 3 U ! Desc R .U/, ! is (pseudo-)functor. LetDesc R .X/ denote the corresponding 2-category fibered in R-linear 2-categories overCov.X/ with object pairs .U; A/ with U 2 Cov.X/ and .U; A/ 2 Desc R .U/; amorphism .U 0 ; A 0 / ! .U; A/ in Desc R .X/ is a pair .; /, where W U 0 ! U is amorphism in Cov.X/ and W .U 0 ; A 0 / ! .U; A/ D .U 0 ; A /.4.3 Trivializations4.3.1 Definition of a trivializationDefinition 4.9. A trivialization of an algebroid stack C on X is an object in C.X/.Suppose that C is an algebroid stack on X and L 2 C.X/ is a trivialization. Theobject L determines a morphism End C .L/ C ! C.Lemma 4.10. The induced morphism D EndC .L/ C ! C is an equivalence.Remark 4.11. Suppose that C is an R-algebroid stack on X. Then, there exists a coverU of X such that the stack 0 C admits a trivialization.4.3.2 The 2-category of trivializations. Let Triv R .X/ denote the 2-category with• objects: the triples .C; U;L/ where C is an R-algebroid stack on X, U is anopen cover of X such that 0 C.N 0U/ is nonempty and L is a trivialization of 0 C,• 1-morphisms: .C 0 ; U 0 ;L/! .C; U;L/are pairs .F; / where W U 0 ! U is amorphism of covers, F W C 0 ! C is a functor such that .N 0 / F.L 0 / D L,• 2-morphisms .F; / ! .G; /, where .F; /; .G; /W .C 0 ; U 0 ;L/! .C; U;L/:the morphisms of functors F ! G.