Mathematics Lecture Notes

Theorem2.7.1Existence and uniqueness of stackification

Let $(\mathcal{C}, \tau)$ be a site and let $\mathcal{F}$ be a category fibered in groupoids over $\mathcal{C}$ . There exists a stack $\mathcal{F}^{\mathrm{st}}$ over $(\mathcal{C}, \tau)$ and a morphism of CFGs

$\eta : \mathcal{F} \longrightarrow \mathcal{F}^{\mathrm{st}}$

satisfying the following universal property: for every stack $\mathcal{G}$ over $(\mathcal{C}, \tau)$ and every morphism $F : \mathcal{F} \to \mathcal{G}$ of CFGs, there exists a morphism of stacks $\tilde{F} : \mathcal{F}^{\mathrm{st}} \to \mathcal{G}$ , unique up to unique 2-isomorphism, such that $\tilde{F} \circ \eta \cong F$ .

Moreover, the pair $(\mathcal{F}^{\mathrm{st}}, \eta)$ is unique up to equivalence: if $(\mathcal{G}, \eta')$ is another such pair, there exists an equivalence $\mathcal{F}^{\mathrm{st}} \simeq \mathcal{G}$ compatible with $\eta$ and $\eta'$ .

Theorem2.7.2Properties of eta

The stackification morphism $\eta : \mathcal{F} \to \mathcal{F}^{\mathrm{st}}$ satisfies:

(i) Locally essentially surjective: For every $U \in \mathcal{C}$ and every object $x \in \mathcal{F}^{\mathrm{st}}(U)$ , there exists a covering $\{U_i \to U\}$ such that $x|_{U_i}$ is in the essential image of $\eta(U_i) : \mathcal{F}(U_i) \to \mathcal{F}^{\mathrm{st}}(U_i)$ for each $i$ .

(ii) Locally fully faithful: For every $U \in \mathcal{C}$ and objects $x, y \in \mathcal{F}(U)$ , the morphism $\eta$ induces an isomorphism of sheaves: $\operatorname{Isom}_\mathcal{F}(x, y)^+ \xrightarrow{\sim} \operatorname{Isom}_{\mathcal{F}^{\mathrm{st}}}(\eta(x), \eta(y))$ where $\operatorname{Isom}_\mathcal{F}(x, y)^+$ denotes the sheafification of the presheaf $\operatorname{Isom}_\mathcal{F}(x, y)$ .

(iii) Characterization: A morphism $\eta : \mathcal{F} \to \mathcal{G}$ of CFGs with $\mathcal{G}$ a stack is a stackification if and only if it satisfies both (i) and (ii).

RemarkAnalogies with sheafification

Properties (i) and (ii) are direct analogues of the properties of sheafification $\theta : F \to F^+$ for presheaves:

(i) is the analogue of " $F^+$ has the same local sections as $F$ ."
(ii) is the analogue of " $F^+$ has the same stalks as $F$ ."

In the presheaf case, $\theta_p : F_p \to F^+_p$ is an isomorphism on stalks. In the stack case, $\eta$ induces isomorphisms on sheafified Isom (the "stacky stalks").

Theorem2.7.3Uniqueness of stackification

Let $\eta : \mathcal{F} \to \mathcal{F}^{\mathrm{st}}$ and $\eta' : \mathcal{F} \to \mathcal{G}$ be two stackifications of $\mathcal{F}$ (both satisfying the universal property). Then there exists an equivalence of stacks $\Psi : \mathcal{F}^{\mathrm{st}} \xrightarrow{\sim} \mathcal{G}$ and a 2-isomorphism $\alpha : \Psi \circ \eta \xRightarrow{\sim} \eta'$ . Moreover, the pair $(\Psi, \alpha)$ is unique up to unique 2-isomorphism.

Proof

By the universal property of $\eta$ , there exists $\tilde{\eta}' : \mathcal{F}^{\mathrm{st}} \to \mathcal{G}$ with $\tilde{\eta}' \circ \eta \cong \eta'$ . By the universal property of $\eta'$ , there exists $\widetilde{\eta} : \mathcal{G} \to \mathcal{F}^{\mathrm{st}}$ with $\widetilde{\eta} \circ \eta' \cong \eta$ . The composition $\widetilde{\eta} \circ \tilde{\eta}' : \mathcal{F}^{\mathrm{st}} \to \mathcal{F}^{\mathrm{st}}$ satisfies $(\widetilde{\eta} \circ \tilde{\eta}') \circ \eta \cong \eta$ . By uniqueness (up to unique 2-isomorphism) in the universal property of $\eta$ , $\widetilde{\eta} \circ \tilde{\eta}' \cong \operatorname{id}_{\mathcal{F}^{\mathrm{st}}}$ . Similarly $\tilde{\eta}' \circ \widetilde{\eta} \cong \operatorname{id}_{\mathcal{G}}$ . So $\tilde{\eta}'$ is an equivalence.

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ExampleStackification of a stack is itself

If $\mathcal{F}$ is already a stack, then $\eta = \operatorname{id} : \mathcal{F} \to \mathcal{F}$ is a stackification. By uniqueness, $\mathcal{F}^{\mathrm{st}} \simeq \mathcal{F}$ .

ExampleStackification reduces to sheafification for sets

Let $F : \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ be a presheaf, viewed as a CFG with discrete fibers. The stackification $F^{\mathrm{st}}$ is equivalent (as a CFG with discrete fibers) to the sheafification $F^+$ . This is because:

For discrete groupoids, "equivalence of categories" reduces to "bijection of sets."
The descent data category for a discrete CFG is a set (discrete groupoid).
The stackification universal property reduces to the sheafification universal property.

ExampleStackification of a presheaf of groupoids

Let $F : \mathcal{C}^{\mathrm{op}} \to \mathbf{Grpd}$ be a presheaf of groupoids (strict functor, giving a split fibered category). The stackification produces a stack $F^{\mathrm{st}}$ that may be non-split.

For example, let $F(U) = BG(U)$ be the presheaf sending every $U$ to the groupoid with one object and automorphism group $G(U)$ . This is not a stack in general (descent fails). The stackification is the classifying stack $BG$ (torsors, not just trivial torsors).

ExampleStackification of orbit presheaf to quotient stack

Let $G$ act on $X$ . The orbit presheaf $\mathcal{O}$ with $\mathcal{O}(U) = X(U)/G(U)$ (as a set, viewed as discrete groupoid) is a CFG. Its stackification is not the quotient stack $[X/G]$ directly, because we lost the automorphism information.

Instead, consider the CFG $\mathcal{F}$ with $\mathcal{F}(U) =$ the groupoid whose objects are elements of $X(U)$ and where $\operatorname{Hom}(x, y) = \{g \in G(U) : g \cdot x = y\}$ . This is the action groupoid $[X(U)/G(U)]$ . The stackification of $\mathcal{F}$ is the quotient stack $[X/G]$ .

ExampleStackification across topologies

Let $G$ be a smooth group scheme over $S$ . Define $BG^{\mathrm{Zar}}$ as the CFG classifying Zariski-locally trivial $G$ -torsors. This is a stack for the Zariski topology but typically not for the etale topology (there are etale-locally trivial torsors that are not Zariski-locally trivial).

The stackification of $BG^{\mathrm{Zar}}$ on the etale site is $BG^{\mathrm{et}}$ (etale-locally trivial torsors). Similarly, the stackification of $BG^{\mathrm{et}}$ on the fppf site is $BG^{\mathrm{fppf}}$ .

For $G = GL_n$ : all torsors are Zariski-locally trivial (Hilbert's Theorem 90), so $BGL_n^{\mathrm{Zar}} = BGL_n^{\mathrm{et}} = BGL_n^{\mathrm{fppf}}$ .

For $G = PGL_n$ : there exist etale-locally trivial but not Zariski-locally trivial $PGL_n$ -torsors (these correspond to Brauer-Severi varieties). So $BPGL_n^{\mathrm{Zar}} \subsetneq BPGL_n^{\mathrm{et}}$ .

ExampleGerbes via stackification

Let $\mathcal{C} = \mathbf{Sch}/S$ with the etale topology, and let $\alpha \in \check{H}^2(S, \mathbb{G}_m)$ be a Brauer class represented by a Cech 2-cocycle $(\alpha_{ijk}) \in \mathbb{G}_m(U_{ijk})$ for a covering $\{U_i \to S\}$ .

Define a CFG $\mathcal{F}_\alpha$ by taking $\mathcal{F}_\alpha(T) = \emptyset$ for $T \to S$ that does not factor through any $U_i$ , and using the cocycle data on the cover. The stackification of $\mathcal{F}_\alpha$ is a $\mathbb{G}_m$ -gerbe over $S$ representing the class $\alpha$ .

The isomorphism class of the gerbe in $H^2(S, \mathbb{G}_m)$ depends only on the cohomology class of $\alpha$ , not the specific cocycle.

Theorem2.7.4Two-step stackification

The stackification can be constructed in two steps:

Step 1 (Prestackification): Define $\mathcal{F}^{\mathrm{ps}}$ with the same objects as $\mathcal{F}$ but with morphisms sheafified: $\operatorname{Hom}_{\mathcal{F}^{\mathrm{ps}}(U)}(x, y) = \operatorname{Isom}_\mathcal{F}(x, y)^+(U)$ Then $\mathcal{F}^{\mathrm{ps}}$ is a prestack with $\eta_1 : \mathcal{F} \to \mathcal{F}^{\mathrm{ps}}$ .

Step 2 (Effectivization): Define $\mathcal{F}^{\mathrm{st}} = (\mathcal{F}^{\mathrm{ps}})^{\mathrm{eff}}$ where objects over $U$ are descent data for $\mathcal{F}^{\mathrm{ps}}$ relative to coverings of $U$ , and morphisms are compatible families of isomorphisms on a common refinement. Then $\mathcal{F}^{\mathrm{st}}$ is a stack with $\eta_2 : \mathcal{F}^{\mathrm{ps}} \to \mathcal{F}^{\mathrm{st}}$ .

The composition $\eta = \eta_2 \circ \eta_1 : \mathcal{F} \to \mathcal{F}^{\mathrm{st}}$ is the stackification.

ExampleTwo-step construction for BG

Step 1: Start with the CFG $\mathcal{F}$ where $\mathcal{F}(S)$ has one object (the "trivial torsor") with $\operatorname{Aut} = G(S)$ . The presheaf $\operatorname{Isom}$ is already a sheaf (it is the sheaf represented by $G$ ), so $\mathcal{F}^{\mathrm{ps}} = \mathcal{F}$ and Step 1 is trivial.

Step 2: Objects of $\mathcal{F}^{\mathrm{st}}(S)$ are descent data: coverings $\{S_i \to S\}$ with trivial torsors on each $S_i$ and transition functions $g_{ij} \in G(S_i \times_S S_j)$ satisfying the cocycle condition. This is precisely a $G$ -torsor on $S$ !

So $\mathcal{F}^{\mathrm{st}} = BG$ , as expected.

ExampleStep 1 matters: non-separated prestacks

Consider the presheaf $F$ on $\mathbb{R}$ (with the usual topology) sending $U$ to the set of bounded continuous functions $U \to \mathbb{R}$ , viewed as a CFG with discrete fibers. This is not a separated presheaf (hence not a prestack): the function $f(x) = x$ is bounded on each interval $(-n, n)$ but not globally, so locality fails.

Step 1: Sheafify Isom (which for a presheaf of sets just means sheafify the presheaf). The prestackification is the sheaf of all continuous functions.

Step 2: Since the prestackification is already a sheaf (= stack with discrete fibers), Step 2 is trivial.

Theorem2.7.5Functoriality of stackification

The stackification construction is functorial: a morphism $F : \mathcal{F} \to \mathcal{G}$ of CFGs induces a morphism $F^{\mathrm{st}} : \mathcal{F}^{\mathrm{st}} \to \mathcal{G}^{\mathrm{st}}$ of stacks, unique up to unique 2-isomorphism, making the diagram $\mathcal{F} \xrightarrow{F} \mathcal{G}$ $\downarrow \eta_\mathcal{F} \quad\quad \downarrow \eta_\mathcal{G}$ $\mathcal{F}^{\mathrm{st}} \xrightarrow{F^{\mathrm{st}}} \mathcal{G}^{\mathrm{st}}$ 2-commutative.

Moreover, a 2-morphism $\alpha : F \Rightarrow G$ between morphisms $\mathcal{F} \to \mathcal{G}$ induces a 2-morphism $\alpha^{\mathrm{st}} : F^{\mathrm{st}} \Rightarrow G^{\mathrm{st}}$ .

In other words, stackification defines a 2-functor $\mathbf{CFG}(\mathcal{C}) \to \mathbf{Stacks}(\mathcal{C})$ that is left 2-adjoint to the inclusion $\mathbf{Stacks}(\mathcal{C}) \hookrightarrow \mathbf{CFG}(\mathcal{C})$ .

ExampleFunctorial application: quotient stacks

Let $\phi : G \to H$ be a homomorphism of group schemes, and let $G$ act on $X$ and $H$ act on $Y$ , with a $\phi$ -equivariant morphism $f : X \to Y$ . Then $f$ induces a morphism of the corresponding action groupoid CFGs, and stackification gives a morphism of quotient stacks: $[X/G] \to [Y/H]$

For instance, the inclusion $G \hookrightarrow G$ (identity) and the action map $G \to *$ give $[G/G] \to [*/G] = BG$ : the inertia map.

RemarkSummary of stackification existence

The stackification theorem is a cornerstone of stack theory:

Existence: Every CFG has a stackification, constructed in two steps (sheafify Isom, then add descent data).
Uniqueness: The stackification is unique up to canonical equivalence.
Universal property: Left 2-adjoint to the inclusion of stacks into CFGs.
Local nature: The stackification does not change local data -- it only enforces global descent.
Functoriality: The construction is functorial in the CFG.

The complete proof is given in the proof section, following the two-step approach.

Math Notes

Stackification Existence

Main Theorem

Properties of the Stackification Morphism

Uniqueness

Examples

The Two-Step Construction

Functoriality

Summary