an scheme and a 1-morphism of fibered categories which is surjective and smooth depends on defining a smooth and surjective morphisms of fibered categories. Here is the algebraic stack from the representable functor on upgraded to a category fibered in groupoids where the categories only have trivial morphisms. This means the setis considered as a category, denoted , with objects in as morphismsand morphisms are the identity morphism. Henceis a 2-functor of groupoids. Showing this 2-functor is a sheaf is the content of the 2-Yoneda lemma. Using the Grothendieck construction, there is an associated category fibered in groupoids denoted . Representable morphisms of categories fibered in groupoids To say this morphism is smooth or surjective, we have to introduce representable morphisms. A morphism of categories fibered in groupoids over is said to be representable if given an object in and an object the 2-fibered product is representable by a scheme. Then, we can say the morphism of categories fibered in groupoids is smooth and surjective if the associated morphismof schemes is smooth and surjective. Deligne-Mumford stacks Algebraic stacks, also known as Artin stacks, are by definition equipped with a smooth surjective atlas , where is the stack associated to some scheme . If the atlas is moreover étale, then is said to be a Deligne-Mumford stack. The subclass of Deligne-Mumford stacks is useful because it provides the correct setting for many natural stacks considered, such as the moduli stack of algebraic curves. In addition, they are strict enough that object represented by points in Deligne-Mumford stacks do not have infinitesimal automorphisms. This is very important because infinitesimal automorphisms make studying the deformation theory of Artin stacks very difficult. For example, the deformation theory of the Artin stack , the moduli stack of rank vector bundles, has infinitesimal automorphisms controlled partially by the Lie algebra . This leads to an infinite sequence of deformations and obstructions in general, which is one of the motivations for studying moduli of stable bundles. Only in the special case of the deformation theory of line bundles is the deformation theory tractable, since the associated Lie algebra is abelian. Note that many stacks cannot be naturally represented as Deligne-Mumford stacks because it only allows for finite covers, or, algebraic stacks with finite covers. Note that because every Etale cover is flat and locally of finite presentation, algebraic stacks defined with the fppf-topology subsume this theory; but,