Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 22 / 3 / c / Solution

The right-adjoint criterion using comma-category colimits is as follows. Under the given colimit-preservation hypothesis, with preservation including the possibly large colimits below, $F:\mathcal{X}\to \mathcal{A}$ has a right adjoint exactly when, for every $A$, the projectionhas a colimit in $\mathcal{X}$. The right adjoint is obtained from these comma-category colimits. Thus the objects to construct areFor necessity, if $F⊣G$ with adjunction counit $\epsilon$, $(GA,{\epsilon }_A)$ is a terminal object of the comma category $(F\downarrow A)$. Its unique incoming morphisms give a colimit cocone for $U_A$, exactly as for the elements projection in the preceding part.

For sufficiency, choose such a colimit $L$ with legs $u_{X,f}:X\to L$. The morphisms $f:FX\to A$ are a cocone on $F{U}_A$. Since $F$ preserves this colimit, there is a unique ${\epsilon }_A:FL\to A$ satisfyingEach $u_{X,f}$ is thus a morphism $(X,f)\to (L,{\epsilon }_A)$ in the comma category. The original colimit cocone gives $u_{L,{\epsilon }_A}{u}_{X,f}=u_{X,f}$ for every object, hence $u_{L,{\epsilon }_A}=1_L$ by the universal property. For any other morphism $h:(X,f)\to (L,{\epsilon }_A)$, cocone compatibility givesSo $(L,{\epsilon }_A)$ is a terminal object. Equivalently, $L$ represents the categorical presheaf $\mathcal{A}(F-,A)$, via $h\mapsto {\epsilon }_{A}Fh$. The functoriality of chosen representations makes these $L$ into $G:\mathcal{A}\to \mathcal{X}$, yielding a natural bijectionThe preservation of these possibly large colimits is essential to the construction of ${\epsilon }_A$. Under a small-only interpretation, the preceding ordinal counterexample also disproves the unqualified converse here. Take $F=X^{\mathrm{op}}:\mathcal{C}\to {\mathbf{Set}}^{\mathrm{op}}$. It preserves all small colimits. For a nonempty set $A$, the comma category $(F\downarrow A)$ has one object over each ordinal and none over $\infty$, so its projection has colimit $\infty$. For $A=\varnothing$, the projection is the identity of $\mathcal{C}$, again with colimit $\infty$. Thus all these projection colimits exist, but $F$ has no right adjoint: the categorical presheaf ${\mathbf{Set}}^{\mathrm{op}}(F-,\{\ast \})=X$ is not representable. This makes the large-preservation qualification substantive.