This section is the continuation of Quot, Section 99.15. Let $\mathcal{C}\! \mathit{urves}$ be the stack whose category of sections over a scheme $S$ is the category of families of curves over $S$. It is somewhat important to keep in mind that a family of curves is a morphism $f : X \to S$ where $X$ is an algebraic space (!) and $f$ is flat, proper, of finite presentation and of relative dimension $\leq 1$. We already know that $\mathcal{C}\! \mathit{urves}$ is an algebraic stack over $\mathbf{Z}$, see Quot, Theorem 99.15.11. If we did not allow algebraic spaces in the definition of our stack, then this theorem would be false.
Often base change is denoted by a subscript, but we cannot use this notation for $\mathcal{C}\! \mathit{urves}$ because $\mathcal{C}\! \mathit{urves}_ S$ is our notation for the fibre category over $S$. This is why in Quot, Remark 99.15.5 we used $B\text{-}\mathcal{C}\! \mathit{urves}$ for the base change
to the algebraic space $B$. The product on the right is over the final object, i.e., over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. The object on the left is the stack classifying families of curves on the category of schemes over $B$. In particular, if $k$ is a field, then
is the moduli stack classifying families of curves on the category of schemes over $k$. Before we continue, here is a sanity check.
Lemma 109.3.1. Let $T \to B$ be a morphism of algebraic spaces. The category is the category of families of curves over $T$.
\[ \mathop{\mathrm{Mor}}\nolimits _ B(T, B\text{-}\mathcal{C}\! \mathit{urves}) = \mathop{\mathrm{Mor}}\nolimits (T, \mathcal{C}\! \mathit{urves}) \]
Proof. A family of curves over $T$ is a morphism $f : X \to T$ of algebraic spaces, which is flat, proper, of finite presentation, and has relative dimension $\leq 1$ (Morphisms of Spaces, Definition 67.33.2). This is exactly the same as the definition in Quot, Situation 99.15.1 except that $T$ the base is allowed to be an algebraic space. Our default base category for algebraic stacks/spaces is the category of schemes, hence the lemma does not follow immediately from the definitions. Having said this, we encourage the reader to skip the proof.
By the product description of $B\text{-}\mathcal{C}\! \mathit{urves}$ given above, it suffices to prove the lemma in the absolute case. Choose a scheme $U$ and a surjective étale morphism $p : U \to T$. Let $R = U \times _ T U$ with projections $s, t : R \to U$.
Let $v : T \to \mathcal{C}\! \mathit{urves}$ be a morphism. Then $v \circ p$ corresponds to a family of curves $X_ U \to U$. The canonical $2$-morphism $v \circ p \circ t \to v \circ p \circ s$ is an isomorphism $\varphi : X_ U \times _{U, s} R \to X_ U \times _{U, t} R$. This isomorphism satisfies the cocycle condition on $R \times _{s, t} R$. By Bootstrap, Lemma 80.11.3 we obtain a morphism of algebraic spaces $X \to T$ whose pullback to $U$ is equal to $X_ U$ compatible with $\varphi $. Since $\{ U \to T\} $ is an étale covering, we see that $X \to T$ is flat, proper, of finite presentation by Descent on Spaces, Lemmas 74.11.13, 74.11.19, and 74.11.12. Also $X \to T$ has relative dimension $\leq 1$ because this is an étale local property. Hence $X \to T$ is a family of curves over $T$.
Conversely, let $X \to T$ be a family of curves. Then the base change $X_ U$ determines a morphism $w : U \to \mathcal{C}\! \mathit{urves}$ and the canonical isomorphism $X_ U \times _{U, s} R \to X_ U \times _{U, t} R$ determines a $2$-arrow $w \circ s \to w \circ t$ satisfying the cocycle condition. Thus a morphism $v : T = [U/R] \to \mathcal{C}\! \mathit{urves}$ by the universal property of the quotient $[U/R]$, see Groupoids in Spaces, Lemma 78.23.2. (Actually, it is much easier in this case to go back to before we introduced our abuse of language and direct construct the functor $\mathit{Sch}/T \to \mathcal{C}\! \mathit{urves}$ which “is” the morphism $T \to \mathcal{C}\! \mathit{urves}$.)
We omit the verification that the constructions given above extend to morphisms between objects and are mutually quasi-inverse. $\square$
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