The Stacks project

Example 93.10.1 (Morphisms of schemes). Let $\mathcal{F}$ be the category defined as follows

  1. an object is a pair $(A, X \to Y)$ consisting of an object $A$ of $\mathcal{C}_\Lambda $ and a morphism $X \to Y$ of schemes over $A$ with both $X$ and $Y$ flat over $A$, and

  2. a morphism $(f, g, h) : (A', X' \to Y') \to (A, X \to Y)$ consists of a morphism $f : A' \to A$ in $\mathcal{C}_\Lambda $ together with morphisms of schemes $g : X \to X'$ and $h : Y \to Y'$ such that

    \[ \xymatrix{ X \ar[r]_ g \ar[d] & X' \ar[d] \\ Y \ar[r]_ h \ar[d] & Y' \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r]^ f & \mathop{\mathrm{Spec}}(A') } \]

    is a commutative diagram of schemes where both squares are cartesian.

The functor $p : \mathcal{F} \to \mathcal{C}_\Lambda $ sends $(A, X \to Y)$ to $A$ and $(f, g, h)$ to $f$. It is clear that $p$ is cofibred in groupoids. Given a morphism of schemes $X \to Y$ over $k$, let $x_0 = (k, X \to Y)$ be the corresponding object of $\mathcal{F}(k)$. We set

\[ \mathcal{D}\! \mathit{ef}_{X \to Y} = \mathcal{F}_{x_0} \]

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View Example 93.10.1 as pdf