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*}

4.4 Products of pairs

Definition 4.4.1. Let $x, y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. A product of $x$ and $y$ is an object $x \times y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ together with morphisms $p\in \mathop{Mor}\nolimits _{\mathcal C}(x \times y, x)$ and $q\in \mathop{Mor}\nolimits _{\mathcal C}(x \times y, y)$ such that the following universal property holds: for any $w\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and morphisms $\alpha \in \mathop{Mor}\nolimits _{\mathcal C}(w, x)$ and $\beta \in \mathop{Mor}\nolimits _\mathcal {C}(w, y)$ there is a unique $\gamma \in \mathop{Mor}\nolimits _{\mathcal C}(w, x \times y)$ making the diagram

\[ \xymatrix{ w \ar[rrrd]^\beta \ar@{-->}[rrd]_\gamma \ar[rrdd]_\alpha & & \\ & & x \times y \ar[d]_ p \ar[r]_ q & y \\ & & x & } \]

commute.

If a product exists it is unique up to unique isomorphism. This follows from the Yoneda lemma as the definition requires $x \times y$ to be an object of $\mathcal{C}$ such that

\[ h_{x \times y}(w) = h_ x(w) \times h_ y(w) \]

functorially in $w$. In other words the product $x \times y$ is an object representing the functor $w \mapsto h_ x(w) \times h_ y(w)$.

Definition 4.4.2. We say the category $\mathcal{C}$ has products of pairs of objects if a product $x \times y$ exists for any $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

We use this terminology to distinguish this notion from the notion of “having products” or “having finite products” which usually means something else (in particular it always implies there exists a final object).


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