Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 24 / 1

We use the general adjoint functor theorem in its solution-set form. Let $G:\mathcal{D}\to \mathcal{C}$ be a functor between locally small categories, with $\mathcal{D}$ a complete category. Then $G$ has a left adjoint exactly when it preserves all small categorical limits and, for each $C\in \mathcal{C}$, there is a set-indexed familysuch that every $u:C\to GD$ factors as $Gf{u}_i$ for some $i$ and some $f:D_i\to D$. This is the solution-set condition. The indexing family is a set, whereas the entire comma category $(C\downarrow G)$ can be large.

We first prove the initial-object lemma for complete categories with a weakly initial set. Let $\mathcal{E}$ be locally small and complete, with a weakly initial set $(E_i{)}_{i\in I}$. Its product in a category $W={\prod }_i{E}_i$ is weakly initial: follow a projection by a map from the corresponding $E_i$ to any desired object. The hom-set $\mathcal{E}(W,W)$ is a set, so we can form the simultaneous equalizerThis is a small limit; equivalently, equalize the family of endomorphisms with the family of identity maps into a product of copies of $W$. Every object receives a map from $E$, since it receives one from $W$.

To prove uniqueness, let $a,b:E\rightrightarrows B$, and take their equalizer $j:V\to E$. Weak initiality of $W$ gives $t:W\to V$. Since $ejt$ is an endomorphism of $W$, the defining property of $e$ givesThe monomorphism $e$ cancels, yielding $jte=1_E$. Thus $j$ is both a monomorphism and a split epimorphism, hence an isomorphism. Its equalizer property now implies $a=b$. Therefore $E$ is initial.

Fix $C$. Small categorical limits in $(C\downarrow G)$ are obtained from the limits in $\mathcal{D}$: preservation by $G$ supplies the unique map from $C$ to the image of the limiting object. The comma category is locally small, and the solution family is a weakly initial set in it. The lemma supplies an initial object $(LC,{\eta }_C:C\to GLC)$.

For $k:C\to C^{\prime }$, initiality gives a unique arrow $Lk:LC\to L{C}^{\prime }$ satisfying $G(Lk){\eta }_C={\eta }_{C^{\prime }}k$. Uniqueness proves preservation of identities and composition, so $L$ is a functor. Initiality also gives natural bijectionswhich prove $L⊣G$.

Conversely, if $L⊣G$, the singleton ${\eta }_C:C\to GLC$ is a solution set by the adjunction bijection. To see that $G$ preserves a small limit $D={\lim }_j{D}_j$, use the adjunction and the limit property to obtain, naturally in $C$,This is precisely the universal property of the image limit cone. It proves the necessity of both hypotheses.

Take the category of ordinals in reverse order ${\mathbf{Ord}}^{\mathrm{op}}$, with a unique arrow $\alpha \to \beta$ exactly when $\alpha \ge \beta$, and the unique functorBoth categories are complete categories. A small limit in the ordinal category is the supremum of the ordinals occurring in the diagram; the empty supremum is $0$, its terminal object. The functor $G$ preserves all these categorical limits, and both categories are locally small categories.

A left adjoint would select an initial object of ${\mathbf{Ord}}^{\mathrm{op}}$, meaning an ordinal at least as large as every ordinal. No such object exists. More specifically, any set of candidate ordinals has an upper bound $\gamma$, and none of its members maps to $\gamma +1$. Thus the solution-set condition fails. This example isolates the genuine smallness requirement in the general adjoint functor theorem.

Let $J:\mathbf{CompHaus}\hookrightarrow \mathbf{Top}$ be the inclusion. Both categories are locally small categories. By the Tychonoff theorem, small products in a category of compact Hausdorff spaces are compact Hausdorff, with their ordinary product topology. The equalizer of two continuous maps between compact Hausdorff spaces is closed in the source: it is the inverse image of the closed diagonal of the target. Hence it is again compact Hausdorff. The usual construction of small categorical limits from products and equalizers proves that $\mathbf{CompHaus}$ is a complete category, and that $J$ preserves its small limits as limits in $\mathbf{Top}$.

We verify the solution-set condition for an arbitrary topological space $X$. Put $\kappa =|X|$ and $\theta =2^{2^{\kappa }}$. Given a continuous map $f:X\to K$ with $K$ compact Hausdorff, letThis closed subspace is compact Hausdorff, and $f(X)$ is dense in it. The allowed cardinal estimate therefore gives $|Y|\le \theta$. The map $f$ factors through the continuous map $X\to Y$ and the subspace inclusion $Y\hookrightarrow K$.

Choose a fixed set $S$ of cardinality $\theta$. Consider all compact Hausdorff topologies on subsets of $S$, together with all continuous maps from $X$ into the resulting spaces. There are only a set of such topologies and functions. Every $Y$ above is homeomorphic to one of these spaces after relabelling its underlying set. Consequently every $X\to JK$ factors through a member of this set of maps. This proves the solution-set condition, including $X=\varnothing$.

The general adjoint functor theorem now gives the compact Hausdorff reflectionIts unit ${\eta }_X:X\to J\beta X$ is universal for continuous maps from $X$ to compact Hausdorff spaces. For arbitrary $X$, this is a reflection, and the unit is not required to be a topological embedding.