A Quillen model for classical Morita theory and a tensor categorification of the Brauer group

Let K be a commutative ring. In this article we construct a well-behaved symmetric monoidal Quillen model structure on the category of small K-categories which enhances classical Morita theory. Making use of it, we then obtain the usual categorification of the Brauer group and of its functoriality. Finally, we prove that the (contravariant) corestriction map for finite Galois extensions also lifts to this categorification.