Let $G = \mathbb{Z}/2\mathbb{Z}$ act on $X = \mathbb{A}^1_k$ by $x \mapsto -x$ ($\operatorname{char} k \neq 2$). The naive presheaf quotient sends $U$ to $X(U)/G(U) = \mathcal{O}(U)/\{\pm 1\}$. This is not a sheaf: consider $f \in \mathcal{O}(U)$; the class $[f]$ and $[-f]$ agree on any cover where $f$ has constant sign, but may not be detected globally.

The prestackification makes $\operatorname{Isom}$ into a sheaf but does not yet ensure effective descent. One further step (adding descent data objects) gives the stackification $[\mathbb{A}^1 / (\mathbb{Z}/2\mathbb{Z})]$.

The CFG of quasi-coherent sheaves $\mathbf{QCoh}$ over $\mathbf{Sch}$ with the fppf topology is already a prestack: for any two quasi-coherent sheaves $\mathcal{E}, \mathcal{F}$ on $S$, the presheaf $T \mapsto \operatorname{Hom}_{T}(f^*\mathcal{E}, f^*\mathcal{F})$ is a sheaf (in fact representable by a quasi-coherent sheaf $\mathcal{H}om(\mathcal{E}, \mathcal{F})$). However, $\mathbf{QCoh}$ is also a stack since descent for quasi-coherent sheaves is effective.

When $\mathcal{F}$ is a presheaf of sets (viewed as a CFG with discrete fibers), the stackification reduces to ordinary sheafification. Step 1 is vacuous (Isom is already a sheaf for discrete groupoids: it is either $\emptyset$ or $\{*\}$). Step 2 adds sections that are locally in $\mathcal{F}$ -- exactly the sheafification construction.

Let $G$ be a smooth group scheme. The naive CFG $BG^{\mathrm{Zar}}$ on the Zariski site classifies Zariski-locally trivial $G$-torsors. When stackified on the etale site, we obtain $BG^{\mathrm{et}}$ classifying etale-locally trivial torsors. The stackification adds new objects: torsors that are only locally trivial in the etale topology.

For $G = GL_n$, every $GL_n$-torsor is already Zariski-locally trivial (by Hilbert 90 / the Zariski triviality of vector bundles on local rings), so $BGL_n^{\mathrm{Zar}} = BGL_n^{\mathrm{et}}$. But for $G = \mu_n$ or $G = PGL_n$, the etale stackification is strictly larger.

Let $G$ act on $X$. The presheaf $U \mapsto X(U)/G(U)$ (orbit presheaf) can be viewed as a CFG. Its stackification is the quotient stack $[X/G]$.

More precisely, consider the CFG $\mathcal{F}$ where $\mathcal{F}(U)$ is the groupoid of "trivial $G$-torsors $P$ over $U$ with $G$-equivariant maps $P \to X$." The objects are just $G$-equivariant maps $G \times U \to X$, i.e., elements of $X(U)$. The stackification adds "twisted" objects: non-trivial torsors with equivariant maps, yielding $[X/G]$.

For a presheaf of sets $F$, the sheafification $F^+$ has the same stalks: $F^+_p \cong F_p$. The analogous statement for stackification is property (3): on "germs" (stalks) and isomorphisms, the stackification does not change the local data. It only changes global information by making descent effective.

For a prestack $\mathcal{F}$ (Axiom 1 already satisfied), the stackification has two equivalent descriptions:

(a) Via descent data): $\mathcal{F}^{\mathrm{st}}(U)$ is the category of descent data for $\mathcal{F}$ relative to all coverings of $U$, modulo refinement.

(b) Via Cech nerve): For a single covering $\pi : V \to U$, form the simplicial object (Cech nerve): $V \times_U V \times_U V \;\substack{\to \\ \to \\ \to}\; V \times_U V \;\substack{\to \\ \to}\; V$ Then $\mathcal{F}^{\mathrm{st}}(U)$ is the 2-limit (homotopy limit) of the cosimplicial diagram obtained by applying $\mathcal{F}$ to the Cech nerve, taken over all coverings.

Consider the presheaf of groupoids $\mathcal{A}z$ where $\mathcal{A}z(S)$ is the groupoid of Azumaya algebras on $S$ (locally free $\mathcal{O}_S$-algebras that are etale-locally matrix algebras). The morphisms are isomorphisms of algebras.

This is already a prestack (Isom is representable), and descent for Azumaya algebras is effective, so $\mathcal{A}z$ is already a stack. However, if we started with "central simple algebras" (only defined over fields), we would need stackification to extend to a stack on $\mathbf{Sch}$.

The isomorphism classes give the Brauer group: $\pi_0(\mathcal{A}z(S)) \cong \operatorname{Br}(S)$ (up to some Morita-equivalence issues). As a stack, $\mathcal{A}z \cong [*/PGL_\infty]$ in a suitable sense.

Consider the morphism of prestacks $F : \mathcal{F} \to \mathcal{G}$ where $\mathcal{G}$ is a stack. By the universal property, $F$ factors uniquely through $\mathcal{F}^{\mathrm{st}}$:

$\mathcal{F} \xrightarrow{\eta} \mathcal{F}^{\mathrm{st}} \xrightarrow{\tilde{F}} \mathcal{G}$

Moreover, $\tilde{F}$ is an equivalence if and only if:

• $F$ is locally essentially surjective: every object of $\mathcal{G}(U)$ is locally in the essential image of $F$.
• $F$ induces isomorphisms on sheafified Isom.

This gives a practical criterion for identifying stackifications.

Let $X$ be a scheme and $G$ a sheaf of abelian groups on $X$. A $G$-gerbe is a stack $\mathcal{G}$ over $X$ that is locally non-empty and locally connected (any two objects are locally isomorphic), with $\operatorname{Aut}(x) \cong G$ for all objects $x$.

Gerbes can be constructed via stackification: start with the "trivial" CFG over $X$ (objects are open subsets, only identity morphisms), then twist by a Cech 2-cocycle with values in $G$. The stackification of this twisted CFG is a $G$-gerbe. The isomorphism class of the gerbe is an element of $H^2(X, G)$.

For $G = \mathbb{G}_m$, we get $\mathbb{G}_m$-gerbes classified by $H^2(X, \mathbb{G}_m) = \operatorname{Br}(X)$ (the cohomological Brauer group).

From the perspective of higher category theory, stackification is a localization: the 2-category of stacks is obtained from the 2-category of CFGs by inverting "local equivalences" (morphisms that are locally essentially surjective and locally fully faithful).

In the language of model categories, CFGs over a site form a model category where:

• Weak equivalences are local equivalences.
• Fibrant objects are stacks.
• Stackification is fibrant replacement.

This perspective connects stackification to the theory of higher stacks and derived algebraic geometry.

Let $X$ be a scheme, $D \subset X$ an effective Cartier divisor, and $r \geq 1$. Consider the CFG $\mathcal{F}$ over $\mathbf{Sch}/X$ where $\mathcal{F}(T \to X)$ is the groupoid of pairs $(\mathcal{L}, \phi)$ with $\mathcal{L}$ a line bundle on $T$ and $\phi : \mathcal{L}^{\otimes r} \to \mathcal{O}_T(D|_T)$ an isomorphism.

This CFG is already a prestack (Isom is representable by $\mu_r$-torsors). The stackification ensures effective descent: locally defined $r$-th roots of $D$ with compatible transition data glue to global $r$-th roots. The resulting stack $\sqrt[r]{D/X}$ is the $r$-th root stack, a Deligne-Mumford stack with $\mu_r$-stabilizers along $D$.

For $X = \mathbb{A}^1$ and $D = \{0\}$ with $r = 2$: the root stack $\sqrt[2]{\{0\}/\mathbb{A}^1}$ looks like $\mathbb{A}^1$ away from the origin but has a $\mu_2$-gerbe at the origin.

The weighted projective stack $\mathcal{P}(a_0, \ldots, a_n)$ is the stackification of the quotient prestack $[\mathbb{A}^{n+1} \setminus \{0\} / \mathbb{G}_m]$ where $\mathbb{G}_m$ acts with weights $(a_0, \ldots, a_n)$. When all weights are 1, the stackification gives $\mathbb{P}^n$ (a scheme). When weights are not all 1, the stackification produces a genuine Deligne-Mumford stack with $\mu_{a_i}$-stabilizers at the coordinate points.

For example, $\mathcal{P}(1, 2)$ is a stack whose coarse moduli space is $\mathbb{P}^1$, but with a $\mu_2$-stacky point at $[0:1]$.