Pullback as Limit
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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 $5.19$