Any set $S$ can be viewed as a discrete groupoid: the objects are elements of $S$, and the only morphisms are identities. The isomorphism classes form the set $S$ itself. A discrete groupoid has trivial automorphism groups.

Any group $G$ defines a groupoid $BG$ with a single object $*$ and $\operatorname{Aut}(*) = G$. This is a connected groupoid with $|BG| = \{*\}$ (one isomorphism class). The groupoid cardinality is $|BG| = 1/|G|$ in the sense of Baez-Dolan.

More generally, any group action $G \curvearrowright S$ defines a translation groupoid $[S/G]$ (or action groupoid): objects are elements of $S$, and $\operatorname{Hom}(s, t) = \{g \in G : g \cdot s = t\}$.

For a topological space $X$, the fundamental groupoid $\Pi_1(X)$ has points of $X$ as objects and homotopy classes of paths as morphisms. The automorphism group of $x \in X$ is $\pi_1(X, x)$. Connected components of $\Pi_1(X)$ correspond to path components of $X$.

A category $\mathcal{F}$ over $\mathcal{C}$ is fibered in groupoids if and only if for every commutative triangle in $\mathcal{C}$: $W \xrightarrow{g} V \xrightarrow{f} U, \quad f \circ g = h$ and every pair of morphisms $\phi : y \to x$ over $f$ and $\psi : z \to x$ over $h$ in $\mathcal{F}$, there exists a unique morphism $\chi : z \to y$ over $g$ with $\phi \circ \chi = \psi$.

This is a very practical criterion: given any diagram in the base and compatible lifts of two of the three arrows, the third lift exists uniquely.

Define $\mathcal{M}_{1,1}$ over $\mathbf{Sch}$: objects over $S$ are elliptic curves $E \to S$ (smooth proper morphisms with geometrically connected fibers of genus 1, equipped with a section $e : S \to E$). Morphisms over $f : T \to S$ are isomorphisms $E_T \cong f^*E$ of elliptic curves over $T$ respecting the sections.

This is fibered in groupoids because:

• Pullback exists by base change.
• Morphisms in each fiber are isomorphisms of elliptic curves.
• The fiber $\mathcal{M}_{1,1}(\operatorname{Spec} k)$ is the groupoid of elliptic curves over $k$, where $\operatorname{Aut}(E) \neq \{1\}$ in general (e.g., $j = 0$ has extra automorphisms in characteristic $\neq 2, 3$).

The fact that $\operatorname{Aut}(E)$ can be nontrivial is precisely why $\mathcal{M}_{1,1}$ is a stack rather than a scheme.

Define $\mathcal{B}un_n$ over $\mathbf{Sch}$: objects over $S$ are vector bundles (locally free sheaves) $\mathcal{E}$ of rank $n$ on $S$. Morphisms from $(T, \mathcal{F})$ to $(S, \mathcal{E})$ over $f : T \to S$ are isomorphisms $\mathcal{F} \cong f^*\mathcal{E}$.

This is fibered in groupoids. The fiber over $S$ is the groupoid of rank-$n$ vector bundles on $S$. The automorphism group of $\mathcal{E}$ is $\operatorname{Aut}(\mathcal{E}) = GL_n(\Gamma(S, \operatorname{End}(\mathcal{E})))$.

For $n = 1$, this is the Picard groupoid, and the set of isomorphism classes is $\operatorname{Pic}(S)$.

The special case $n = 1$ of the previous example: $\mathcal{P}ic$ over $\mathbf{Sch}$ with objects $(S, \mathcal{L})$ where $\mathcal{L}$ is an invertible sheaf on $S$. For any line bundle $\mathcal{L}$, $\operatorname{Aut}(\mathcal{L}) = \mathcal{O}_S(S)^{\times} = \mathbb{G}_m(S)$. So every object has automorphism group $\mathbb{G}_m$.

As a stack, $\mathcal{P}ic \cong B\mathbb{G}_m$, the classifying stack of $\mathbb{G}_m$.

The fibered category $\mathcal{M}_g$ over $\mathbf{Sch}$ parametrizes smooth proper curves of genus $g$. Objects over $S$ are smooth proper morphisms $C \to S$ with geometrically connected fibers of genus $g$. Morphisms are isomorphisms compatible with base change.

• For $g = 0$: every fiber is $\mathbb{P}^1$, but $\operatorname{Aut}(\mathbb{P}^1) = PGL_2$. So $\mathcal{M}_0 \cong BPGL_2$.
• For $g = 1$ (without marked point): $\operatorname{Aut}(E)$ contains the elliptic curve itself (translation), so automorphisms are large.
• For $g \geq 2$: $\operatorname{Aut}(C)$ is a finite group (bounded by $84(g-1)$ by Hurwitz), and is trivial for a "general" curve.

The Deligne-Mumford theorem states that $\mathcal{M}_g$ (for $g \geq 2$) is a smooth proper Deligne-Mumford stack over $\mathbb{Z}$, and its coarse moduli space $M_g$ is a quasi-projective variety.

For an algebraic group $G$ over $S$, the classifying stack $BG$ is the CFG over $\mathbf{Sch}/S$ where $BG(T) = \{\text{principal } G_T\text{-torsors over } T\}$. Morphisms are $G$-equivariant isomorphisms.

Key features:

• $BG(\operatorname{Spec} k) = \{G\text{-torsors over } k\} = H^1(k, G)$ (as a pointed set, first Galois cohomology).
• Every object has $\operatorname{Aut}(P) = G(T)$ when $P$ is trivial (and a form of $G$ in general).
• $BG$ is an Artin stack (algebraic stack in the sense of Artin).

Specific instances:

• $B\mathbb{G}_m$: classifies line bundles. The Picard stack.
• $BGL_n$: classifies rank-$n$ vector bundles.
• $B\mu_n$: classifies $\mu_n$-torsors, related to $n$-th roots of line bundles.
• $B(\mathbb{Z}/2\mathbb{Z})$: classifies etale double covers.

The fibered category $\mathcal{M}_{g,n}$ over $\mathbf{Sch}$ parametrizes smooth curves of genus $g$ with $n$ distinct marked points. Objects over $S$ are tuples $(C \to S, \sigma_1, \ldots, \sigma_n)$ where $\sigma_i : S \to C$ are disjoint sections. Morphisms are isomorphisms respecting base change and marked points.

The condition $2g - 2 + n > 0$ ensures stability (finite automorphisms). For example:

• $\mathcal{M}_{0,3}$: genus 0 with 3 points. $\operatorname{Aut} = 1$ (3 points determine an isomorphism of $\mathbb{P}^1$), so $\mathcal{M}_{0,3} \cong \operatorname{Spec} \mathbb{Z}$.
• $\mathcal{M}_{0,4}$: genus 0 with 4 points. $\mathcal{M}_{0,4} \cong \mathbb{P}^1 \setminus \{0, 1, \infty\}$ (the cross-ratio).
• $\mathcal{M}_{1,1}$: the moduli of elliptic curves, a Deligne-Mumford stack with coarse space the $j$-line.

For $\mathcal{M}_g$, the set of isomorphism classes $\pi_0(\mathcal{M}_g(S))$ is the set of isomorphism classes of smooth genus-$g$ curves over $S$. Taking $S = \operatorname{Spec} k$ for an algebraically closed field $k$, this is the classical moduli set $M_g(k)$.

However, the passage $\mathcal{F}_U \mapsto \pi_0(\mathcal{F}_U)$ loses information about automorphisms. Two CFGs can have the same isomorphism classes but different automorphism groups, hence different stacks. This is why working with the full groupoid-valued functor is essential.

For $BG$ where $G$ is a finite group over a field $k$, the fiber functor sends $S$ to the groupoid of $G$-torsors over $S$. The presheaf of isomorphism classes sends $S$ to $H^1_{\mathrm{et}}(S, G)$.

Over $\operatorname{Spec} k$ with $k$ algebraically closed: every $G$-torsor is trivial, so $\pi_0(BG(k)) = \{*\}$ (a single point). But $BG(k) \simeq BG$ as a groupoid (one object with automorphism group $G$). The automorphism information is invisible at the level of isomorphism classes but crucial for the stack structure.

The morphism $\mathcal{M}_{1,1} \to \mathcal{M}_1$ that sends $(E, e)$ (an elliptic curve with origin) to $E$ (the underlying genus-1 curve, forgetting the marked point) is a morphism of CFGs. On fibers over $S$, this is the functor that forgets the section.

This is not an equivalence because $\mathcal{M}_1$ has "more" automorphisms: the automorphism group of an elliptic curve in $\mathcal{M}_1$ includes all translations, while in $\mathcal{M}_{1,1}$ only origin-preserving automorphisms survive.

The determinant gives a morphism of CFGs: $\det : \mathcal{B}un_n \to \mathcal{P}ic$ sending a rank-$n$ vector bundle $\mathcal{E}$ to its determinant line bundle $\det(\mathcal{E}) = \bigwedge^n \mathcal{E}$. This is compatible with pullback since $\det(f^*\mathcal{E}) \cong f^*\det(\mathcal{E})$.

Taking $F = G = \operatorname{id} : \mathcal{X} \to \mathcal{X}$ and forming the 2-fiber product $\mathcal{X} \times_{\mathcal{X} \times \mathcal{X}} \mathcal{X}$, we get the inertia stack $\mathcal{I}_\mathcal{X}$:

$\mathcal{I}_\mathcal{X}(U) = \{(x, \alpha) : x \in \mathcal{X}(U), \alpha \in \operatorname{Aut}(x)\}$

The inertia stack encodes the automorphism groups of objects. For a scheme $X$ (viewed as a CFG), $\mathcal{I}_X = X$ (all automorphisms are trivial). For $BG$, the inertia stack is $[G/G]$ where $G$ acts on itself by conjugation.

Consider the morphism $\mathcal{M}_{1,1} \to B(\mathbb{Z}/n\mathbb{Z})^2$ that sends an elliptic curve $E$ to its $n$-torsion $E[n]$ (as a torsor). The fiber product $\mathcal{M}_{1,1}(n) = \mathcal{M}_{1,1} \times_{B(\mathbb{Z}/n\mathbb{Z})^2} \operatorname{Spec}\mathbb{Z}[1/n]$ parametrizes elliptic curves with full level-$n$ structure. For $n \geq 3$, the automorphisms are killed and $\mathcal{M}_{1,1}(n)$ is a scheme (the modular curve $Y(n)$).

For $BG$ with $G$ a smooth group scheme, let $P, Q$ be two $G$-torsors over $S$. Then $\operatorname{Isom}(P, Q)$ is the sheaf on $\mathbf{Sch}/S$ sending $T \to S$ to the set of $G$-equivariant isomorphisms $P_T \to Q_T$. This is representable by the scheme $\operatorname{Isom}_G(P, Q) = P \times^G Q^{\mathrm{op}}$ (a twisted form of $G$).

In particular, $\operatorname{Aut}(P) = \operatorname{Isom}(P, P)$ is a form of $G$ (equal to $G$ when $P$ is trivial).