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:

• In $\mathsf{Set}$, the pullback of two set functions $f:A\to C$ and $g:B\to C$ is given by a subset $D\subset A\times B$ of the cartesian product of $A$ and $B$, consisting of the set of pairs $(a,b)$ such that $f(a)=g(b)$, endowed with restrictions of the projection morphisms on $A\times B$, denoted $p_A:D\to A$ and $p_B:D\to B$.

• In $\mathsf{Top}$, consider the diagram $\ast\rightarrow S^1\leftarrow \mathbb R$, where the former morphism sends the single point in the one-pointed space $\ast$ to any point in $S^1$, and the latter morphism "wraps" $\mathbb R$ infinitely many times around $S^1$ counterclockwise. The pullback of this diagram is given by the discrete space $\mathbb Z$, with the trivial morphism $\mathbb Z\rightarrow\ast$ and the inclusion map $\mathbb Z\rightarrow\mathbb R$.

• In $\mathsf{CRing}$, consider the diagram $2\mathbb Z\rightarrow \mathbb Z\leftarrow 3\mathbb Z$ where the arrows are inclusion maps. The pullback is given by $6\mathbb Z$ with inclusion maps $\rightarrow 2\mathbb Z$ and $\rightarrow 3\mathbb Z$.