A Lie 2-group G is a category internal to the category of Lie groups. Consequently it is a monoidal category and a Lie groupoid. The Lie groupoid structure on G gives rise to the Lie 2-algebra X(G) of multiplicative vector fields. The monoidal structure on G gives rise to a left action of the 2-group G on the Lie groupoid G, hence to an action of G on the Lie 2-algebra X(G). As a result we get the Lie 2-algebra X(G)^G of left-invariant multiplicative vector fields.
On the other hand there is a well-known construction that associates a Lie 2-algebra g to a Lie 2-group G: apply the functor Lie : LieGp -> LieAlg to the structure maps of the category G. We show that the Lie 2-algebra g is isomorphic to the Lie 2-algebra X(G)^G of left invariant multiplicative vector fields.
Keywords: Lie 2-group, Lie 2-algebra, invariant vector fields, 2 limit
2010 MSC: 18D05, 22A22, 22E, 17B
Theory and Applications of Categories, Vol. 34, 2019, No. 21, pp 604-634.