In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves and the moduli stack of elliptic curves. Originally, they were introduced by Grothendieck to keep track of automorphisms on moduli spaces, a technique which allows for treating these moduli spaces as if their underlying schemes or algebraic spaces are smooth. But, through many generalizations the notion of algebraic stacks was finally discovered by Michael Artin. Definition Motivation One of the motivating examples of an algebraic stack is to consider a groupoid scheme over a fixed scheme . For example, if (where is the group scheme of roots of unity), , is the projection map, is the group actionand is the multiplication mapon . Then, given an -scheme , the groupoid scheme forms a groupoid (where are their associated functors). Moreover, this construction is functorial on forming a contravariant 2-functorwhere is the 2-category of small categories. Another way to view this is as a fibred category through the Grothendieck construction. Getting the correct technical conditions, such as the Grothendieck topology on , gives the definition of an algebraic stack. For instance, in the associated groupoid of -points for a field , over the origin object there is the groupoid of automorphisms . Note that in order to get an algebraic stack from , and not just a stack, there are additional technical hypotheses required for . Algebraic stacks It turns out using the fppf-topology (faithfully flat and locally of finite presentation) on , denoted , forms the basis for defining algebraic stacks. Then, an algebraic stack is a fibered categorysuch that is a category fibered in groupoids, meaning the overcategory for some is a groupoid The diagonal map of fibered categories is representable as algebraic spaces There exists an scheme and an associated 1-morphism of fibered categories which is surjective and smooth called an atlas. Explanation of technical conditions Using the fppf topology First of all, the fppf-topology is used because it behaves well with respect to descent. For example, if there are schemes and can be refined to an fppf-cover of , if is flat, locally finite type, or locally of finite presentation, then has this property. this kind of idea