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

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Sources

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