For $\mathcal{C} = \mathbf{Sch}$ (or $\mathbf{Sch}/S$) with the fppf topology, every scheme $X$ defines a representable stack $h_X$ with $h_X(T) = \operatorname{Hom}(T, X)$ (discrete groupoid). The etale and fppf topologies are subcanonical, so representable presheaves are automatically sheaves.

For example, $h_{\mathbb{A}^n}(T) = \mathcal{O}(T)^n$ (the set of $n$-tuples of global functions on $T$).

The representable CFG $h_{\operatorname{Spec} A}$ for a ring $A$ sends $T = \operatorname{Spec} B$ to $\operatorname{Hom}(A, B)$ (ring homomorphisms). For $A = \mathbb{Z}[x]$, we get $h_{\mathbb{A}^1}(T) = \mathcal{O}(T)$ (the "generic point"). For $A = \mathbb{Z}[x, y]/(xy - 1)$, we get $h_{\mathbb{G}_m}(T) = \mathcal{O}(T)^*$ (units).

Let $\mathcal{F} = \mathcal{M}_g$ (moduli of genus-$g$ curves) over $\mathbf{Sch}$. By the 2-Yoneda lemma:

$\operatorname{Hom}(h_S, \mathcal{M}_g) \simeq \mathcal{M}_g(S) = \{\text{smooth proper curves } C \to S \text{ of genus } g\}$

So a morphism $S \to \mathcal{M}_g$ (i.e., $h_S \to \mathcal{M}_g$) is the same as a family of curves over $S$. The "universal family" corresponds to the identity morphism $\mathcal{M}_g \to \mathcal{M}_g$ (though $\mathcal{M}_g$ is not representable).

By the 2-Yoneda lemma: $\operatorname{Hom}(h_S, BG) \simeq BG(S) = \{\text{principal } G\text{-torsors over } S\}$

A morphism $S \to BG$ corresponds to a $G$-torsor on $S$. The trivial morphism (composition $S \to * \to BG$) corresponds to the trivial torsor $G \times S$.

For $G = \mathbb{G}_m$: morphisms $S \to B\mathbb{G}_m$ correspond to line bundles on $S$. The "tautological" line bundle on $B\mathbb{G}_m$ corresponds to the identity map $B\mathbb{G}_m \to B\mathbb{G}_m$.

For $k$ a field and $\mathcal{X}$ a stack over $\mathbf{Sch}/k$: $\mathcal{X}(k) := \operatorname{Hom}(h_{\operatorname{Spec} k}, \mathcal{X}) \simeq \mathcal{X}(\operatorname{Spec} k)$

This is a groupoid, not a set. The "set of $k$-points" is $|\mathcal{X}(k)| = \pi_0(\mathcal{X}(k))$ (isomorphism classes), but the full groupoid contains automorphism information.

For $\mathcal{X} = B\mathbb{G}_m$ over $k$ algebraically closed: $|\mathcal{X}(k)| = \{*\}$ (one point), but $\operatorname{Aut}(*) = k^*$. For $\mathcal{X} = \mathcal{M}_2$: $|\mathcal{X}(k)|$ parametrizes genus-2 curves up to isomorphism.

Given two morphisms $f, g : h_S \to \mathcal{F}$ corresponding to objects $x, y \in \mathcal{F}(S)$, a 2-morphism $\alpha : f \Rightarrow g$ corresponds to an isomorphism $x \xrightarrow{\sim} y$ in $\mathcal{F}(S)$.

In particular, the 2-automorphisms of a morphism $f : h_S \to \mathcal{F}$ correspond to $\operatorname{Aut}_{\mathcal{F}(S)}(x)$. This is how stacks encode automorphism groups: the group of self-2-morphisms of a point $S \to \mathcal{X}$.

Let $P \to S$ be a $G$-torsor. The corresponding morphism $f : h_S \to BG$ is such that the 2-fiber product $h_S \times_{BG} h_S$ is representable by the scheme $P$ (viewed as a $G$-scheme over $S$):

$h_S \times_{BG} h_S \simeq h_P$

More precisely, a $T$-point of $h_S \times_{BG} h_S$ consists of two maps $T \to S$ with an isomorphism of the corresponding torsors, which is the same as a section of $P_T$, which by definition is a map $T \to P$.

This shows that the diagonal $\Delta : h_S \to h_S \times_{BG} h_S$ is a $G$-torsor, and hence $BG$ has "affine diagonal" when $G$ is affine.

The forgetful morphism $\mathcal{M}_{g,1} \to \mathcal{M}_g$ (forget the marked point) is representable. Given a family $C \to S$ (a morphism $h_S \to \mathcal{M}_g$), the fiber product is:

$\mathcal{M}_{g,1} \times_{\mathcal{M}_g} h_S \simeq h_C$

A $T$-point of the fiber product is a family $C_T \to T$ together with a section $T \to C_T$, which is the same as a map $T \to C$ over $S$. So the fiber product is the total space $C$ itself. This is the universal curve over $\mathcal{M}_g$ (in a relative sense).

For a stack $\mathcal{X}$, the diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable if and only if for any two morphisms $h_U \to \mathcal{X}$ and $h_V \to \mathcal{X}$ (corresponding to objects $x \in \mathcal{X}(U)$ and $y \in \mathcal{X}(V)$), the Isom presheaf $\operatorname{Isom}(x, y)$ is representable by an object (scheme, algebraic space) of $\mathcal{C}$.

This is a key condition in the definition of algebraic stacks:

• Deligne-Mumford stack: diagonal is representable, unramified, and separated.
• Artin stack: diagonal is representable and of finite type.

The algebraic group $GL_n$ has functor of points $GL_n(S) = GL_n(\mathcal{O}(S))$ for affine $S = \operatorname{Spec} R$. The classifying stack $BGL_n$ has:

$BGL_n(S) = \{\text{rank-}n \text{ vector bundles on } S\}$

A morphism $f : S \to BGL_n$ is the same as a rank-$n$ vector bundle on $S$, by the 2-Yoneda lemma.

The quotient stack $[\operatorname{Spec} k / GL_n]$ has: $[\operatorname{Spec} k / GL_n](S) = \{(P, \sigma) : P \text{ a } GL_n\text{-torsor on } S, \sigma : P \to \operatorname{Spec} k \text{ equivariant}\} \simeq BGL_n(S)$ since every equivariant map to a point is unique.

The Hilbert scheme $\operatorname{Hilb}^n_{X/S}$ represents the functor: $T \mapsto \{Z \subset X_T : Z \to T \text{ flat, proper, with Hilbert polynomial } n\}$

By Yoneda (classical, since this is a scheme), a morphism $T \to \operatorname{Hilb}^n$ is the same as a flat family of subschemes parametrized by $T$. The 2-Yoneda lemma tells us the analogous story for stacks.

For $\mu_n = \operatorname{Spec} k[t]/(t^n - 1)$, the classifying stack $B\mu_n$ parametrizes $\mu_n$-torsors:

$B\mu_n(S) = \{(\mathcal{L}, \phi) : \mathcal{L} \text{ a line bundle on } S, \phi : \mathcal{L}^{\otimes n} \xrightarrow{\sim} \mathcal{O}_S\}$

An object is a line bundle with a trivialization of its $n$-th tensor power -- an "$n$-th root of the trivial bundle." The automorphism group is $\mu_n(S)$ acting by $\zeta \cdot (\mathcal{L}, \phi) = (\mathcal{L}, \zeta^n \phi)$ (but $\zeta^n = 1$, so $\operatorname{Aut} = \mu_n$).

The 2-Yoneda embedding $\mathbf{Sch} \hookrightarrow \mathbf{Stacks}$ is 2-fully faithful. This means:

• Schemes form a full sub-2-category of stacks.
• A morphism between representable stacks $h_X \to h_Y$ is the same as a morphism of schemes $X \to Y$.
• There are no nontrivial 2-morphisms between morphisms of schemes (since the fibers are discrete).

A stack $\mathcal{X}$ is representable (equivalent to $h_X$ for some scheme $X$) if and only if $\mathcal{X}(S)$ is a discrete groupoid (set) for all $S$ and $\mathcal{X}$ is a sheaf.

The 2-Yoneda lemma gives a criterion: a stack $\mathcal{X}$ is representable if and only if:

1. All automorphism groups $\operatorname{Aut}(x)$ are trivial for all $x \in \mathcal{X}(S)$.
2. The presheaf of isomorphism classes $S \mapsto \pi_0(\mathcal{X}(S))$ is a sheaf.
3. This sheaf is representable.

Condition (1) is the key distinction: stacks with nontrivial automorphisms can never be schemes.

For $\mathcal{M}_g$ ($g \geq 2$): condition (1) fails (the general curve has trivial automorphisms, but special curves like hyperelliptic curves have $\mathbb{Z}/2\mathbb{Z}$). So $\mathcal{M}_g$ is not a scheme.