The 2-group of linear auto-equivalences of an abelian category and its Lie 2-algebra

For any abelian category \calC satsifying (AB5) over a separated, quasi-compact scheme S, we construct a stack of 2-groups \GL(\calC) over the flat site of S. We will give a concrete description of \GL(\calC) when \calC is the category of quasi-coherent sheaves on a separated, quasi-compact scheme X over S. We will show that the tangent space \gl(\calC) of \GL(\calC) at the origin has a structure as a Lie 2-algebra.