Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 75 / 6

Identify the reflective subcategory with its full replete image, writing ${\eta }_X:X\to LX$ for the reflection unit. An object $Z$ is fixed by $L$ exactly when precomposition with every ${\eta }_X$ gives a bijection $\mathcal{E}(LX,Z)\to \mathcal{E}(X,Z)$. One implication is the reflection adjunction; for the other, take $X=Z$ to obtain a retraction of ${\eta }_Z$, and uniqueness of maps into the fixed object $LZ$ makes it a two-sided inverse.

An ambient finite limit of fixed objects has this same property: its hom-sets are limits of the hom-sets of the diagram, and a limit of these bijections is a bijection. Thus the ambient terminal object, products and equalizers are fixed and retain their universal properties in the full subcategory. The reflective subcategory is Cartesian, with finite limits computed in the ambient topos.

Let $A,B$ be fixed objects and form the ambient exponential object $B^A$. For every $X$, the exponential adjunction identifies maps $X\to B^A$ with maps $X\times A\to B$. Since $L$ is left exact and $LA=A$,and its unit corresponds to ${\eta }_X\times 1_A$. Because $B$ is fixed, every map $X\times A\to B$ therefore extends uniquely to $LX\times A\to B$. Transposing back givesThe fixed-object criterion from part (i) shows that $B^A$ already belongs to $\mathcal{L}$. The ambient evaluation and transpose universal property restrict to the full subcategory, proving the subcategory is cartesian closed. This constructs exponentials in a left-exact reflective subcategory, rather than merely reflecting an exponential and assuming its universal property survives.

For a subobject $m:S\hookrightarrow X$, define the closure operation induced by a left-exact reflector bywhere the pullback uses ${\eta }_X:X\to LX$ and the monomorphism $Lm:LS\hookrightarrow LX$. Naturality of the unit factors $S\hookrightarrow X$ through $c_X(S)$, proving extensivity, and functoriality of $L$ proves monotonicity. Applying the left-exact $L$ to the defining pullback gives $L(c_X(S))\cong LS$, since $L{\eta }_X$ is invertible. Repeating the construction therefore changes nothing, proving idempotence.

For $h:Y\to X$, naturality of $\eta$ and preservation of pullbacks identify the closure of $h^{\ast }S$ with $Y{\times }_{LX}LS$, which is also $h^{\ast }{c}_X(S)$. HenceThe operation is thus universal: it is stable under every pullback. The same construction preserves finite intersections because $L$ is left exact.

Every characteristic map $a:X\to \Omega$ expresses its subobject as $a^{\ast }\top$. Universality of the closure operation induced by a left-exact reflector givesBy definition, $j$ classifies $c_{\Omega }(\top )$, so the right-hand subobject has characteristic map $j\circ a$. The original and its closure agree exactly when their unique characteristic maps agree. ThusExtensivity, idempotence, preservation of intersections and closure of the whole subobject also give the Lawvere-Tierney topology laws for $j$.

Let ${\Omega }_j\hookrightarrow \Omega$ be the equalizer of $j$ and the identity. By part (iv), maps $X\to {\Omega }_j$ correspond exactly to closed subobjects of $X$. Closed subobjects of $X$ in turn correspond to subobjects of $LX$ in $\mathcal{L}$: send $S$ to $LS\hookrightarrow LX$, and in the reverse direction pull a fixed subobject $N\hookrightarrow LX$ back along ${\eta }_X$. Left exactness shows that these operations are inverse; the pulled-back object has reflection $N$, making it closed by the defining formula.

ConsequentlyReplacing $X$ by $LX$ and using $L{\eta }_X$ invertible shows that maps into ${\Omega }_j$ extend uniquely across every unit. The criterion of part (i) therefore proves ${\Omega }_j$ itself is fixed, so belongs to $\mathcal{L}$. Truth factors through it because $j\top =\top$. For fixed $X$, the displayed bijection becomesWith the factored truth arrow this is precisely the subobject classifier universal property. Thus the closed-subobject classifier supplies the classifier in the reflective subcategory.