Category theory; homological algebra Definition

Pushout

Definition

Let $\mathcal{C}$ be a category, and let $f \colon Z \to X$ and $g \colon Z \to Y$ be morphisms of $\mathcal{C}$ . A pushout of $f$ and $g$ is an object $P$ with morphisms $i_1 \colon X \to P$ and $i_2 \colon Y \to P$ such that:

$i_1 \circ f = i_2 \circ g$ .
For every object $Q$ and morphisms $j_1 \colon X \to Q$ and $j_2 \colon Y \to Q$ with $j_1 \circ f = j_2 \circ g$ , there is a unique morphism $u \colon P \to Q$ with $u \circ i_1 = j_1$ and $u \circ i_2 = j_2$ .

Remarks

Equivalently, $(P, i_1, i_2)$ is a colimit of the diagram $X \xleftarrow{f} Z \xrightarrow{g} Y$ . A pushout is unique up to a unique isomorphism, and one writes $P = X \sqcup_Z Y$ or $X +_Z Y$ .

Example: in the category of sets, let $\sim$ be the smallest equivalence relation on $X \sqcup Y$ with $f(z) \sim g(z)$ for all $z \in Z$ . Let $P = (X \sqcup Y)/{\sim}$ , and let $i_1, i_2$ send an element to its class. Clause 1 holds by the choice of the relation. For clause 2, the map $u$ must send the class of $x$ to $j_1(x)$ and the class of $y$ to $j_2(y)$ ; this is well defined because $j_1 \circ f = j_2 \circ g$ .

Non-example: take $P = X \sqcup Y$ with the two inclusions and no identification. If $Z$ is not empty, then $i_1(f(z))$ and $i_2(g(z))$ lie in different parts, so clause 1 fails.

In the category of groups the pushout is the amalgamated free product $X *_Z Y$ .

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Category concept
Colimit concept
Morphism concept
Pushout definition

Tag	`Z3MQ1D5Q38`
URI	`m2://Z3MQ1D5Q38`
Kind	definition
Writing	W0 stub
MSC	18A30
Defines	Pushout
Uses	Category, Morphism, Colimit

Pushout

Definition

Remarks

Reading path

Further reading

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