Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 22 / 5

Write $K:\mathcal{B}\to \mathcal{C}$, $J:\mathcal{C}\to \mathcal{B}$, with adjunction unit $\rho$ and adjunction counit $\epsilon$. The adjunction givesFor $f:C\to C^{\prime }$, precomposition with ${\epsilon }_C$ followed by this bijection givesby the triangle identities for an adjunction. Thus $J$ is fully faithful exactly when precomposition with ${\epsilon }_C$ is bijective for every $C,C^{\prime }$.

A morphism $e:P\to Q$ with this property is an isomorphism. Surjectivity for target $P$ supplies $u:Q\to P$ with $ue=1_P$. Since $(eu)e=e=1_{Q}e$, injectivity for target $Q$ gives $eu=1_Q$. Conversely, precomposition with an isomorphism is always bijective. Applied to every ${\epsilon }_C$, this proves the fully faithful right-adjoint criterion:When the adjunction counit is invertible, the inverse to $f\mapsto Jf$ is explicitly $h\mapsto {\epsilon }_{C^{\prime }}K(h){\epsilon }_C^{-1}$.

If $B$ belongs to the full subcategory $\mathcal{C}$, the adjunction and fullness of its inclusion give, for every $B^{\prime }\in \mathcal{B}$,Therefore membership implies the hom-set condition. In addition, $J$ is fully faithful, so the adjunction counit is invertible by the preceding criterion. The triangular equation $J{\epsilon }_B{\rho }_B=1_B$ then shows that ${\rho }_B$ is an isomorphism. As in the printed inclusion notation, $J$ is suppressed when writing $KB$ as an object of $\mathcal{B}$.

Assume the hom-set condition. Its surjectivity at $B^{\prime }=B$ gives $u:KB\to B$ with $u{\rho }_B=1_B$. We show that this splitting is two-sided.

Because $J$ is fully faithful, ${\epsilon }_{KB}$ is invertible. Both triangle identities for an adjunction give, after suppressing $J$,Applying $K$ to $u{\rho }_B=1_B$ yields $Ku{\rho }_{KB}=1_{KB}$. Naturality of the adjunction unit at $u$ gives ${\rho }_{B}u=Ku{\rho }_{KB}$, hence ${\rho }_{B}u=1_{KB}$. Consequently the hom-set condition forces the unit to be invertible:Only surjectivity at $B^{\prime }=B$ was needed for this direction; the given family of bijections certainly supplies it.

If ${\rho }_B:B\to KB$ is an isomorphism, then $B$ is isomorphic in $\mathcal{B}$ to an object of $\mathcal{C}$. Since $\mathcal{C}$ is a replete subcategory, it contains $B$. Thus invertibility of the unit implies membership, completing the cycle of implications:

Let $J:\mathcal{C}\hookrightarrow \mathcal{B}$ exhibit a reflective subcategory, with reflector $K$, and let $E:I\to \mathcal{C}$ be a small diagram in a category. Completeness of $\mathcal{B}$ gives a categorical limit $L$ of $JE$. For every $B^{\prime }\in \mathcal{B}$, the universal property of this categorical limit and the reflection adjunction giveThe composite is precomposition with ${\rho }_{B^{\prime }}$, so $L$ satisfies the hom-set condition from the preceding part. Its proof of invertibility of ${\rho }_L$ did not require repleteness. Thus $L\cong JKL$ whether or not the chosen full reflective subcategory is replete.

Transport the ambient limit cone along ${\rho }_L^{-1}:JKL\to L$. Its legs lie in the full subcategory, and their ambient universal property, restricted to objects of $\mathcal{C}$, is exactly the internal categorical limit property. Hence every small diagram in the reflective subcategory has a limit:For a replete subcategory, the ambient limit object $L$ itself belongs to $\mathcal{C}$. This argument includes the empty diagram and requires no limit-preservation hypothesis on the reflector.