Moduli of linear and abelian categories

Linear categories naturally have several identification relations : isomorphisms, categorical equivalences and Morita equivalences. In this thesis, we construct the classifying stacks for these three relations ($\ukcatiso$, $\ukcateq$, $\ukcatmor$) together with the classifying stack of abelian categories ($\ukab$), the originality of the subject being that, apart from the first one, these are higher stacks. The principal result is that, under some finiteness assumptions, these stacks are geometric in the sense of C. Simpson. In particular, one recover the Hochschild cohomology of a category $C$ as the tangent complex, i.e. the object classifying first order deformations of $C$, of these stacks at the point defined by $C$. Moreover, there exists a natural sequence of surjective morphisms of stacks : $$\ukcatiso \tto \ukcateq \tto \ukcatmor \tto \ukab$$ for which we prove that the middle one is etale, and the right one is an equivalence.