Pullback as Equalizer

Theorem

Let $\mathbf C$ be a metacategory with binary products.

Suppose that the following diagram commutes (possibly except for the square):

$\quad\quad \begin{xy}\xymatrix@+1em{ P \ar@/^/[rrd]^*+{p_1} \ar@/_/[ddr]_*+{p_2} \ar[rd]^*+{e} \\ & A \times B \ar[r]_*+{\pi_1} \ar[d]^*+{\pi_2} & A \ar[d]^*+{f} \\ & B \ar[r]_*+{g} & C }\end{xy}$

where $A \times B$ is the binary product of $A$ and $B$, and $\pi_1, \pi_2$ are the associated projections.

Then the border of the diagram, i.e.:

$\quad\quad \begin{xy}\xymatrix{ P \ar[r]^*+{p_1} \ar[d]_*+{p_2} & A \ar[d]^*+{f} \\ B \ar[r]_*+{g} & C }\end{xy}$

is a pullback of $f$ and $g$ if and only if $e: P \to A \times B$ is the equalizer of $f \circ \pi_1$ and $g \circ \pi_2$.

Corollary

Suppose $\mathbf C$ has all binary products and all equalizers.

Then $\mathbf C$ has all pullbacks.

Proof

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Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 5.2$: Proposition $5.5$