The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 4.32.2. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category over $\mathcal{C}$.

  1. The composition of two strongly cartesian morphisms is strongly cartesian.

  2. Any isomorphism of $\mathcal{S}$ is strongly cartesian.

  3. Any strongly cartesian morphism $\varphi $ such that $p(\varphi )$ is an isomorphism, is an isomorphism.

Proof. Proof of (1). Let $\varphi : y \to x$ and $\psi : z \to y$ be strongly cartesian. Let $t$ be an arbitrary object of $\mathcal{S}$. Then we have

\begin{align*} & \mathop{Mor}\nolimits _\mathcal {S}(t, z) \\ & = \mathop{Mor}\nolimits _\mathcal {S}(t, y) \times _{\mathop{Mor}\nolimits _\mathcal {C}(p(t), p(y))} \mathop{Mor}\nolimits _\mathcal {C}(p(t), p(z)) \\ & = \mathop{Mor}\nolimits _\mathcal {S}(t, x) \times _{\mathop{Mor}\nolimits _\mathcal {C}(p(t), p(x))} \mathop{Mor}\nolimits _\mathcal {C}(p(t), p(y)) \times _{\mathop{Mor}\nolimits _\mathcal {C}(p(t), p(y))} \mathop{Mor}\nolimits _\mathcal {C}(p(t), p(z)) \\ & = \mathop{Mor}\nolimits _\mathcal {S}(t, x) \times _{\mathop{Mor}\nolimits _\mathcal {C}(p(t), p(x))} \mathop{Mor}\nolimits _\mathcal {C}(p(t), p(z)) \end{align*}

hence $z \to x$ is strongly cartesian.

Proof of (2). Let $y \to x$ be an isomorphism. Then $p(y) \to p(x)$ is an isomorphism too. Hence $\mathop{Mor}\nolimits _\mathcal {C}(p(z), p(y)) \to \mathop{Mor}\nolimits _\mathcal {C}(p(z), p(x))$ is a bijection. Hence $\mathop{Mor}\nolimits _\mathcal {S}(z, x) \times _{\mathop{Mor}\nolimits _\mathcal {C}(p(z), p(x))} \mathop{Mor}\nolimits _\mathcal {C}(p(z), p(y))$ is bijective to $\mathop{Mor}\nolimits _\mathcal {S}(z, x)$. Hence the displayed map of Definition 4.32.1 is a bijection as $y \to x$ is an isomorphism, and we conclude that $y \to x$ is strongly cartesian.

Proof of (3). Assume $\varphi : y \to x$ is strongly cartesian with $p(\varphi ) : p(y) \to p(x)$ an isomorphism. Applying the definition with $z = x$ shows that $(\text{id}_ x, p(\varphi )^{-1})$ comes from a unique morphism $\chi : x \to y$. We omit the verification that $\chi $ is the inverse of $\varphi $. $\square$


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