higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
An algebraic stack is essentially a geometric stack on the étale site.
Depending on details, this is a Deligne-Mumford stack or a more general Artin stack in the traditional setup of algebraic spaces.
Let $C_{fppf}$ be the fppf-site and $\mathcal{E} = Sh_{(2,1)}(C_{fppf})$ the (2,1)-topos of stacks over it.
An algebraic stack is
an object $\mathcal{X}\in Sh_{(2,1)}(C_{fppf})$;
such that
the diagonal $\mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces;
there exists a scheme $U \in Sh(C_{fppf}) \hookrightarrow Sh_{(2,1)}(C_{fppf})$ and a morphism $U \to \mathcal{X}$ which is a surjective and smooth morphism.
This appears in this form as (deJong, def. 47.12.1).
A smooth algebraic groupoid is an internal groupoid in algebraic spaces such that source and target maps are smooth morphisms.
This appears as (deJong, def. 47.16.2).
Notice that every internal groupoid in algebraic spaces represents a (2,1)-presheaf on the fppf-site. We shall not distinguish between the groupoid and the stackification of this presheaf, called the quotient stack of the groupoid.
Every algebraic stack is equivalent to a smooth algebraic groupoid and every smooth algebraic groupoid is an algebraic stack.
This appears as (deJong, lemma 47.16.2, theorem 47.17.3).
Orbifolds are an example of an Artin stack. For orbifolds the stabilizer groups are finite groups, while for Artin stacks in general they are algebraic groups.
A noncommutative generalization for Q-categories instead of Grothendieck topologies, hence applicable in noncommutative geometry of Deligne–Mumford and Artin stacks can be found in (KontsevichRosenberg).
algebraic stack
A standard textbook reference is
An account is given in chapter 47 of
Course notes are in
A brief overview is in
The noncommutative version is discussed in