Pullback as Limit

Theorem

Let $\mathbf C$ be a metacategory.

Let $f_1: A \to C$ and $f_2: B \to C$ be morphisms of $\mathbf C$.

Let their pullback:

$\begin{xy}\xymatrix@+1em@L+3px{ P \ar[r]^*+{p_2} \ar[d]_*+{p_1} & A \ar[d]^*+{f_1} \\ B \ar[r]_*+{f_2} & C }\end{xy}$

exist in $\mathbf C$.

Then $\struct {P, p_1, p_2}$ is the limit of the diagram $D: \mathbf J \to \mathbf C$ defined by:

$\begin{xy}\xymatrix@+1em@L+3px{ \save[]+<0em,-2em>*{\mathbf{J}:} \restore & & \cdot \ar[d] \\ & \cdot \ar[r] & \cdot }\end{xy}$

$\begin{xy}\xymatrix@+1em@L+3px{ \save[]+<0em,-2em>*{D:} \restore & & A \ar[d]^*+{f_1} \\ & B \ar[r]_*+{f_2} & C }\end{xy}$

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 5.4$: Example $5.19$