The Stacks project

Definition 4.6.1. Let $x, y, z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $f\in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$ and $g\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(z, y)$. A fibre product of $f$ and $g$ is an object $x \times _ y z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ together with morphisms $p \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x \times _ y z, x)$ and $q \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x \times _ y z, z)$ making the diagram

\[ \xymatrix{ x \times _ y z \ar[r]_ q \ar[d]_ p & z \ar[d]^ g \\ x \ar[r]^ f & y } \]

commute, and such that the following universal property holds: for any $w\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and morphisms $\alpha \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(w, x)$ and $\beta \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(w, z)$ with $f \circ \alpha = g \circ \beta $ there is a unique $\gamma \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(w, x \times _ y z)$ making the diagram

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

commute.

Comments (2)

Comment #155 by on

In the lower right corner of the second diagram, replace by .

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  • 4 comment(s) on Section 4.6: Fibre products

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View Definition 4.6.1 as pdf