Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 22 / 4

The general adjoint functor theorem has the following limit form. Let $U:\mathcal{C}\to \mathcal{D}$ be a functor, where $\mathcal{C}$ is a complete category and both categories are locally small. Then $U$ has a left adjoint exactly when it preserves small limits and satisfies the solution-set condition.

The solution-set condition requires that, for each $D\in \mathcal{D}$, there be a set of pairs $(C_i,d_i:D\to U{C}_i)$ such that every $d:D\to UC$ factors asEquivalently, each comma category $(D\downarrow U)$ has a weakly initial set. All completeness and preservation requirements here concern small categorical limits. Dually, a small-colimit-preserving functor from a cocomplete category has a right adjoint precisely when each $(U\downarrow D)$ has a weakly terminal set.

The Special adjoint functor theorem says: if $\mathcal{C}$ is a locally small category, a complete category, a well-powered category, and has a small cogenerating family, then a functor $U:\mathcal{C}\to \mathcal{D}$ into any locally small category has a left adjoint if and only if it preserves small categorical limits.

A small cogenerating family $(Q_i{)}_{i\in I}$, with $I$ a set, distinguishes unequal parallel morphisms by postcomposition: for $f\ne g:A\to B$, some $q:B\to Q_i$ has $qf\ne qg$. Being well-powered means that the subobjects of each object form a set up to the usual equivalence of monomorphisms. These hypotheses make the solution-set condition automatic.

The dual formulation uses a cocomplete category, being a well-copowered category and having a small generating family, and concludes that every small-colimit-preserving functor has a right adjoint.

It is enough to prove the solution-set condition and then apply the general adjoint functor theorem. Fix $D\in \mathcal{D}$ and a morphism $d:D\to UC$. We will factor it through one object in a set depending only on $D$.

First construct a minimal supported subobject. Among the subobjects $m:M\hookrightarrow C$ for which $d=U(m)d_M$ for some $d_M:D\to UM$, include $1_C$ and take their intersection $m_0:C_0\hookrightarrow C$. This is a small intersection, by well-poweredness. It exists by completeness as the categorical limit of the diagram consisting of these monomorphisms into $C$. Its map to $C$ is a monomorphism: two maps with the same composite to $C$ have equal projections to every $M$, since each $m$ is monic, and are then equal by the categorical limit property. One can equivalently construct these intersections by pullbacks in a category and small products in a category, using stability of monomorphisms under pullback in a category.

Choose the factorizations $d_M$. They form a compatible categorical cone into the image diagram under $U$, all with common composite $d$ to $UC$. Preservation of small categorical limits yields $d_0:D\to U{C}_0$ with $U(m_0)d_0=d$. If $n:N\hookrightarrow C_0$ is another subobject through which $d_0$ factors after applying $U$, then $m_{0}n$ is among the original supported subobjects. The intersection property gives $r:C_0\to N$ with $m_{0}nr=m_0$. Since $m_0$ is monic, $nr=1_{C_0}$; since $n$ is monic, also $rn=1_N$. Thus every supported subobject of $C_0$ is invertible.

For each member $Q_i$ of the small cogenerating family, considerThis map is injective. If $U(q)d_0=U(q^{\prime })d_0$, preservation of the equalizer of $q,q^{\prime }$ makes $d_0$ factor through the image of that equalizer. Minimality makes its inclusion an isomorphism, so $q=q^{\prime }$.

Write $R_i=\mathcal{D}(D,U{Q}_i)$; it is a set by local smallness of $\mathcal{D}$. Let $S_i\subseteq R_i$ be the image of ${\theta }_i$. For each $s\in S_i$, there is exactly one corresponding $q_{i,s}:C_0\to Q_i$. These maps define the evaluation embedding into cogenerator productsIt is a monomorphism: if $eg=eh$ and $g\ne h$, cogeneration supplies some $q:C_0\to Q_i$ distinguishing $g,h$; that $q$ is one of the projections of $e$, a contradiction. The product is small. It is important to use the subfamilies $S_i$, since some missing coordinate in $R_i$ need not correspond to a morphism out of $C_0$.

There are only a set of possible families $(S_i\subseteq R_i{)}_{i\in I}$. For each such family form $P_S$, choose a set of representatives $N\hookrightarrow P_S$ of its subobjects, and take all pairs $(N,d_N:D\to UN)$. Local smallness of $\mathcal{D}$ and well-poweredness make their union a set. In the case just constructed, $e$ identifies $C_0$ with one chosen representative $N$. Transporting $d_0$ to $UN$ and composing the inverse identification with $m_0$ gives a factorization of the original $d$ through that pair. Therefore these pairs form a weakly initial set in $(D\downarrow U)$.

The general adjoint functor theorem now supplies a left adjoint to $U$. Conversely, a right adjoint preserves small categorical limits, either by the adjunction hom-set bijections and the hom-set detection of categorical limits, or directly from their universal properties. Hence the stated special theorem is proved in both directions.