Category:Definitions/Pullbacks

This category contains definitions related to Pullbacks in the context of Category Theory.
Related results can be found in Category:Pullbacks.

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 $f^* g$.