Could not include topos theory - contents
derived smooth geometry
An algebraic stack is
This appears in this form as (deJong, def. 47.12.1).
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).
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).
A standard textbook reference is
An account is given in chapter 47 of
The noncommutative version is discussed in