process

Deformations of gerbes on smooth manifolds 365is a morphism of R-modules 01 W A 01 ! B 01 such that the diagram.pr 2 01 / A 01 ˝R .pr 2 12 / A 01A 012.pr 2 02 / A 01is commutative..pr 2 01 / . 01 /˝.pr 2 12 / . 01 /.pr 2 02 / . 01 / (4.5).pr 2 01 / B 01 ˝R .pr 2 12 / B 012B 01 .pr 2 02 / B 01The 1-morphism induces a morphism of R-algebras WD .pr 0 00 / . 01 /W A ! Bon N 0 U as well as morphisms p iWD .pr p i / ./W A p i! B p ion N p U. The morphism 01 is compatible with the morphism of algebras 0 ˝ op1 W A1 0 ˝R .A 1 1 /op ! B0 1 ˝R.B1 1/op and the respective module structures.A 1-morphism of descent data is a 1-morphism of the underlying convolution datawhich preserves respective units.4.2.4 2-morphisms. Suppose that we are given descent data .U; A/ and .U; B/ asin 4.2.2 and two 1-morphismsas in 4.2.3.A 2-morphism; W .U; A/ ! .U; B/b W !is a section b 2 .N 0 UI B/ such that the diagramA 01b˝ 01 B 1 0 ˝R B 0101˝bB 01 ˝R B11 B 01is commutative.4.2.5 The 2-category of descent data. FixacoverU of X.Suppose that we are given descent data .U; A/, .U; B/, .U; C/ and 1-morphisms W .U; A/ ! .U; B/ and W .U; B/ ! .U; C/. The map 01 ı 01 W A 01 ! C 01 isa 1-morphism of descent data ı W .U; A/ ! .U; C/, the composition of and .Let .i/ W .U; A/ ! .U; B/, i D 1; 2; 3, be1-morphisms and let b .j / W .j / ! .j C1/ , j D 1; 2, be2-morphisms. The section b .2/ b .1/ 2 .N 0 UI B/ defines a2-morphism, denoted b .2/ b .1/ W .1/ ! .3/ , the vertical composition of b .1/ and b .2/ .