# 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:

- $\begin{xy}\xymatrix{ Q \ar[r]^*+{q_1} \ar[d]_*+{q_2} & A \ar[d]^*+{f} \\ B \ar[r]_*+{g} & C }\end{xy}$

- $\begin{xy}\xymatrix@+1em{ Q \ar@/^/[rrd]^*+{q_1} \ar@/_/[ddr]_*+{q_2} \ar@{-->}[rd]^*+{u} \\ & P \ar[r]_*+{p_1} \ar[d]^*+{p_2} & A \ar[d]^*+{f} \\ & B \ar[r]_*+{g} & C }\end{xy}$

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**.

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 5.2$: Definition $5.4$