process

Deformations of gerbes on smooth manifolds 355acts on MC 2 .g/ 1 . 1 ; 2 /. Let exp t 2 exp g 1 and let exp X 2 MC 2 .g/ 1 . 1 ; 2 /.Then.exp t/ .exp X/ D exp.dt C Œ; t/ expX 2 exp g 0 :Such an element exp t is called a 2-morphism between exp X and .exp t/.exp X/.Wedenote by MC 2 .g/ 2 .exp X; exp Y/the set of 2-morphisms between exp X and exp Y .This set is endowed with a vertical composition given by the product in the groupexp g 1 .Let 1 ; 2 ; 3 2 MC 2 .g/ 0 . Let exp X 12 ; exp Y 12 2 MC 2 .g/ 1 . 1 ; 2 / and exp X 23 ,exp Y 23 2 MC 2 .g/ 1 . 2 ; 3 /. Then one defines the horizontal compositionas follows. Let˝W MC 2 .g/ 2 .exp X 23 ; exp Y 23 / MC 2 .g/ 2 .exp X 12 ; exp Y 12 /! MC 2 .g/ 2 .exp X 23 exp X 12 ; exp X 23 exp Y 12 /exp 2t 12 2 MC 2 .g/ 2 .exp X 12 ; exp Y 12 /; exp 3t 23 2 MC 2 .g/ 2 .exp X 23 ; exp Y 23 /:Thenexp 3t 23 ˝ exp 2t 12 D exp 3t 23 exp 3.e ad X 23.t 12 //To summarize, the data described above forms a 2-groupoid which we denote byMC 2 .g/ as follows:1. the set of objects is MC 2 .g/ 0 ,2. the groupoid of morphisms MC 2 .g/. 1 ; 2 /, i 2 MC 2 .g/ 0 consists of• objects, i.e., 1-morphisms in MC 2 .g/ are given by MC 2 .g/ 1 . 1 ; 2 / – thegauge transformations between 1 and 2 ,• morphisms between exp X, expY 2 MC 2 .g/ 1 . 1 ; 2 / which are given byMC 2 .g/ 2 .exp X; exp Y/.A morphism of nilpotent DGLA W g ! h induces a functor W MC 2 .g/ !MC 2 .g/.We have the following important result ([12], [11] and references therein).Theorem 3.1. Suppose that W g ! h is a quasi-isomorphism of DGLA and let m be anilpotent commutative ring. Then the induced map W MC 2 .g ˝ m/ ! MC 2 .h ˝ m/is an equivalence of 2-groupoids.3.2 Deformations and Deligne 2-groupoid. Let k be an algebraically closed field ofcharacteristic zero.