DG coalgebras as formal stacks

Abstract

The category of unital (unbounded) dg cocommutative coalgebras over a field of characteristic zero is provided with a structure of simplicial closed model category. This generalizes the model structure defined by Quillen in 1969 for 2-reduced coalgebras. In our case, the notion of weak equivalence is structly stronger than that of quasi-isomorphism. A pair of adjoint functors connecting the category of coalgebras with the category of dg Lie algebras, induces an equivalence of the corresponding homotopy categories. The model category structure allows one to consider dg coalgebras as the most general formal stacks. The corresponding Lie algebra is then interpreted as a tangent Lie algebra which defines the formal stack uniquely up to a weak equivalence. As an example, we calculate the coalgebra of formal deformations of a principal G-bundle on a scheme X.