Pullback of Subset Inclusion

Theorem

Denote with $\mathbf{Set}$ the category of sets.

Let $A, B$ be sets, and let $f: A \to B$ be a mapping.

Let $V \subseteq B$ be a subset of $B$.

Denote with $i: V \to B$ the inclusion mapping.

Let $f^{-1} \left({V}\right) \subseteq A$ be the preimage of $V$ under $f$.

Denote with $j: f^{-1} \left({V}\right) \to A$ the inclusion mapping.

Denote with $\bar f = f \restriction_{f^{-1} \left({V}\right)}$ the restriction of $f$ to $f^{-1} \left({V}\right)$.

Then:

$\quad\quad \begin{xy}\xymatrix{ f^{-1} \left({V}\right) \ar[r]^*+{\bar f} \ar[d]_*+{j} & V \ar[d]^*+{i} \\ A \ar[r]_*+{f} & B }\end{xy}$

is a pullback diagram in $\mathbf{Set}$.

From the definition of pullback, given a commutative diagram:

$\quad\quad \begin{xy}\xymatrix{ Q \ar[r]^*+{q_1} \ar[d]_*+{q_2} & V \ar[d]^*+{i} \\ A \ar[r]_*+{f} & B }\end{xy}$

we need to find a mapping $u: Q \to \map {f^{-1}} V$ such that $\bar f \circ u = q_1$ and $j \circ u = q_2$.

Let $x \in Q$; then $\map {q_1} x \in V \subseteq B$.

Note that the commutativity of the diagram implies that $\map {q_1} x = \map f {\map {q_2} x}$ since $i$ is an inclusion.

Hence also $\map f {\map {q_2} x} \in V$.

It follows by definition of preimage that $\map {q_2} x \in \map {f^{-1}} V$.

Thus define $u: Q \to \map {f^{-1}} V$ by restriction of $q_2$:

$\map u x := \map {q_2 \restriction_{Q \times \map {f^{-1}} V}} x$

Then we have:

$\map {\paren {j \circ u}} x = \map u x = \map {q_2} x$

Hence $q_2 = j \circ u$ by Equality of Mappings.

Furthermore:

$\map {\bar f \circ u} x = \map {\bar f \circ q_2} x = \map f {\map {q_2} x} = \map {q_1} x$

since $\bar f$ and $u$ are restrictions of $f$ and $q_2$, respectively.

Hence $\bar f \circ u = q_1$ by Equality of Mappings.

Observe that any $u': Q \to \map {f^{-1}} V$ must satisfy:

$j \circ u' = q_2 = j \circ u$

This establishes that $u: Q \to \map {f^{-1}} V$ is unique.

Therefore, we have established that:

$\quad\quad \begin{xy}\xymatrix{ \map {f^{-1}} V \ar[r]^*+{\bar f} \ar[d]_*+{j} & V \ar[d]^*+{i} \\ A \ar[r]_*+{f} & B }\end{xy}$

is a pullback diagram in $\mathbf{Set}$.

$\blacksquare$