Franklin's Notes


Pushout and pullback

A pullback is a limit of the diagram indexed by the abstract category $\cdot\rightarrow\cdot\leftarrow\cdot$. In particular, given objects $A,B,C$ with $f:A\to C$ and $g:B\to C$, a pullback consists of an object $D$ with morphisms $h:D\to A$ and $j:D\to B$ such that the following diagram commutes:

However, the pullback cannot consist of just any $D$, for this would just comprise a cone and not necessarily a limit cone. $P$ (together with its morphisms $\to A$ and $\to B$) comprises a pullback if, for every $D$ making the outer quadrilateral commute, there exists a unique morphism $D\to P$ making the whole diagram commute. In other words, the cone with summit $P$ is a terminal object in the category of cones over the diagram $A\rightarrow C\leftarrow B$. This situation is shown in the diagram below:

A pushout, on the other hand, is a colimit of the diagram indexed by $\cdot\leftarrow\cdot\rightarrow\cdot$. This is a dual concept to the pullback.

Here are some examples of pullbacks and pushouts in various familiar categories:

category-theory

categorical-limits

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