Pullbacks & Pushouts

Intersections, preimages and matching pairs are one construction; turn the arrows around and it becomes gluing.

The idea

The pullback is the universal property shared by three familiar constructions: the intersection of two subsets, the preimage of a subset along a function, and the set of pairs that two given functions send to the same place. Each starts from two functions with a common target and keeps the pairs that agree there.

Written with arrows only, that description becomes a definition.

Definition (Pullback).

Given arrows $f \colon A \to C$ and $g \colon B \to C$ with a common target, a pullback of them is an object $P$ with arrows $q \colon P \to A$ and $p \colon P \to B$ satisfying $f \circ q = g \circ p$, and universal among all such data: whenever an object $T$ carries $u \colon T \to A$ and $v \colon T \to B$ with $f \circ u = g \circ v$, exactly one arrow $\bar{t} \colon T \to P$ has $q \circ \bar{t} = u$ and $p \circ \bar{t} = v$.

In $\mathbf{Set}$ the pullback is the set of pairs $(a,b)$ with $f(a) = g(b)$, and the three constructions above are what that set becomes when one or both of the functions is an inclusion.

Reversing every arrow gives the dual. A pushout starts from two arrows with a common source and asks for a universal object receiving both. In $\mathbf{Set}$ it glues: lay the two sets side by side and identify the two images of each element of the shared object. A pullback cuts a product down by an equation; a pushout builds a disjoint union up by identifications.

Ways to work on it

• Walkthrough. Define pullbacks and pushouts, and recognize them as intersections, preimages, and gluings.
• Proof. Building a pullback out of a product and an equalizer — the first sign that products and equalizers generate everything.
• Practice. Compute small pullbacks and pushouts in familiar categories.
• Hardest. Work through examples where pullbacks and pushouts behave unexpectedly.

Not sure where to start? Take the ten-question placement test.