Section 4.9 (0025): Pushouts—The Stacks project

Definition 4.9.1. Let $x, y, z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $f\in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(y, x)$ and $g\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(y, z)$. A pushout of $f$ and $g$ is an object $x\amalg _ y z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ together with morphisms $p\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x, x\amalg _ y z)$ and $q\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(z, x\amalg _ y z)$ making the diagram

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

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

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

commute.

It is possible and straightforward to prove the uniqueness of the triple $(x\amalg _ y z, p, q)$ up to unique isomorphism (if it exists) by direct arguments. Another possibility is to think of the pushout as the fibre product in the opposite category, thereby getting this uniqueness for free from the discussion in Section 4.6.

Definition 4.9.2. We say a commutative diagram

\[ \xymatrix{ y \ar[r] \ar[d] & z \ar[d] \\ x \ar[r] & w } \]

in a category is cocartesian if $w$ and the morphisms $x \to w$ and $z \to w$ form a pushout of the morphisms $y \to x$ and $y \to z$.

The Stacks project

4.9 Pushouts

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