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If $G$ is an affine groupe scheme over some base $S$, you can consider the groupoid $G\rightrightarrows S$. The corresponding prestack $[G\rightrightarrows S]^{pre}$ is (equivalent to) the prestack of …

About 1. : no, smoothness isn't essential. "Flat is enough" : De Jong's slogan to express this result due to M.Artin. https://www.math.columbia.edu/~dejong/wordpress/?p=1584 I quote : "Given a flat, f …

Besides the references already given, I like Dan Edidin's Notes on the construction of the moduli space of curves https://arxiv.org/abs/math/9805101 I quote : "In section 3 we return to curves and …

As a complement to the answers above : it is kind of well-known (at least I thought it was) that the natural morphism $$\operatorname{\mathbf{Hom}}_{gr} (G,H) \to \operatorname{\mathbf{Hom}}(BG,BH)$ …

A typical example from deformation theory : fix $i:X_0\to X$ of first order thickening defined by a square zero ideal $\mathcal I$, and let $\mathcal E_0$ be a locally free sheaf of finite rank on $X …

You can get a lot of examples by dimension shifting. Namely, consider any exact sequence of groups $$1\to K\to G \to H\to 1 \; .$$ Fix a $H$-torsor $T$. The stack $\mathcal G_T$ of liftings of the str …

A morphism of stacks is an iso iff it is a mono and an epi F. $F'$ is an epi iff $F$ is And this is enough : First case : assume $F$ is representable, i.e. $\Delta_F$ is a mono. …

Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic stacks over $S$. …

over an algebraically closed field of characteristic zero say (but would work in characteristic p as well by defining precisely X as a stack of roots in the sense of Vistoli - see Charles Cadman, Using stacks … Also, you may want to consider the following closely related notion, taken from Fundamental Groups of Algebraic Stacks Behrang Noohi http://arxiv.org/abs/math/0201021 "An algebraic stack being uniformizable …

And fortunately there is an excellent reference online: Tag04Y1 in the Stacks Project. I quote: Lemma 8.4. Let $C$ be a site. …

The notion of "torsor under a groupoid in a topos" already appears in Breen, Lawrence Tannakian categories. Motives (Seattle, WA, 1991), 337–376, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc …