Equalizer as Limit

Theorem

Let $\mathbf C$ be a metacategory.

Let $f_1, f_2: C_1 \to C_2$ be morphisms of $\mathbf C$.

Let their equalizer $e: E \to C_1$ exist in $\mathbf C$.

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

$\begin{xy}\xymatrix@+1em@L+3px{

\mathbf{J}:

&

\ast
 \ar[r]<2pt>
 \ar[r]<-2pt>

&

\star

}\end{xy}$

$\begin{xy}\xymatrix@+1em@L+3px{

D:

&

C_1
 \ar[r]<2pt>^*+{f_1}
 \ar[r]<-2pt>_*+{f_2}

&

C_2

}\end{xy}$

Proof

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Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 5.4$: Example $5.18$