# Definition:Pullback (Category Theory)

## Definition

Let $\mathbf C$ be a metacategory.

Let $f: A \to C$ and $g: B \to C$ be morphisms with common codomain.

A pullback of $f$ and $g$ is a commutative diagram:

$\begin{xy}\xymatrix{ P \ar[r]^*+{p_1} \ar[d]_*+{p_2} & A \ar[d]^*+{f} \\ B \ar[r]_*+{g} & C }\end{xy}$

such that $f \circ p_1 = g \circ p_2$, subject to the following UMP:

For any commutative diagram:
$\begin{xy}\xymatrix{ Q \ar[r]^*+{q_1} \ar[d]_*+{q_2} & A \ar[d]^*+{f} \\ B \ar[r]_*+{g} & C }\end{xy}$
there is a unique morphism $u: Q \to P$ making the following diagram commute:
$\begin{xy}\xymatrix@+1em{ Q \ar@/^/[rrd]^*+{q_1} \ar@/_/[ddr]_*+{q_2} \ar@{-->}[rd]^*+{u} \\ & P \ar[r]_*+{p_1} \ar[d]^*+{p_2} & A \ar[d]^*+{f} \\ & B \ar[r]_*+{g} & C }\end{xy}$

In this situation, $p_1$ is called the pullback of $f$ along $g$ and may be denoted as $g^* f$.

Similarly, $p_2$ is called the pullback of $g$ along $f$ and may be denoted $f^* g$.

## Also known as

In many areas of mathematics, and consequently in many books, $P$ is written as $A \times_C B$.

It is then sometimes called the fibred product of $A$ and $B$ over $C$.

This convention potentially obfuscates $f$ and $g$, and so is not encouraged.

Depending on the context, $P$ may also be denoted as $g^*A$ and called the pullback of $A$ along $g$.

Naturally enough, it is sometimes also referred to as $f^*B$ and called the pullback of $B$ along $f$.

This vocabulary is usually employed when one of $f$ or $g$ is implicit from what $B$ or $A$ represents, respectively.

Pullback diagrams are also sometimes called Cartesian squares.

## Also see

• Results about pullbacks can be found here.