  We prove that the category of dg-modules and dg-algebras in a Grothendieck
quasi-abelian category are endowed with a Quillen model structure. This allows
some flexibility in setting up a theory of derived algebraic geometry in the
infinite dimensional setting. For example, the category of complete
bornological vector spaces, or equivalently, convenient vector spaces, is a
Grothendieck quasi-abelian category. Closely related is the Grothendieck
quasi-abelian category of ind-Banach spaces whose associated model category is
shown to be Quillen equivalent. Applications include the Chevalley-Eilenberg
resolution and the Koszul resolution of a commutative monoid object in a
Grothendieck quasi-abelian category. These can be used for the calculation of
derived quotients by an infinite dimensional Lie algebra and derived
intersections respectively.
