Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 17 / 2

Let $\hat{\mathcal{O}}=[\mathcal{O}(S{)}^{\mathrm{op}},\mathbf{Set}]$, with categorical presheaf restrictions $X(U)\to X(V)$ when $V\subseteq U$. The five functors areIn $\Delta A$ all restriction maps are identities. In $\Lambda A$ and $\nabla A$, all nonidentity restrictions are the unique maps permitted by the displayed sets. These definitions are valid also when $A$ is empty: a restriction out of $(\nabla A)(S)$ still has singleton target. Maps of sets act at the displayed $A$-components, and natural transformations act on $\Pi$ and $\Gamma$ by evaluation.

A natural transformation $\Lambda A\to X$ is determined by an arbitrary map $A\to X(\varnothing )$, since all other source components are empty. A natural transformation $X\to \Delta A$ is determined by its empty-open component $X(\varnothing )\to A$: the component at $U$ must first restrict $X(U)\to X(\varnothing )$. Similarly, a natural transformation $\Delta A\to X$ is determined by $A\to X(S)$, followed by the restriction $X(S)\to X(U)$. Finally, a natural transformation $X\to \nabla A$ is determined by $X(S)\to A$, since all proper-open targets are singletons. These descriptions give the hom-set bijectionsThus the five adjoints for constant presheaves on open sets areThe values at the empty open set are genuine presheaf values here; no sheaf condition or sheafification is being imposed.

Let $C\mathcal{A}={\pi }_0\mathcal{A}$ be the set of connected components of a category: two objects of a category are identified when joined by a finite zigzag of morphisms. Let $DA$ be the discrete category on $A$, and let $IA$ be the indiscrete category on $A$, with exactly one morphism for every ordered pair of objects.

A functor $\mathcal{A}\to DA$ must give the same object value along every arrow, hence along every zigzag. Conversely a function ${\pi }_0\mathcal{A}\to A$ uniquely defines such a functor. A functor $DA\to \mathcal{A}$ is just a choice of an object of a category for each element of $A$. A functor $\mathcal{A}\to IA$ is likewise determined by an arbitrary object function, since its arrow images are forced and always compose correctly. ThereforeThis is the adjoint chain for the objects of a category:

Use the preservation obstructions to extending an adjoint chain: a functor having a further left adjoint must preserve categorical limits, while one having a further right adjoint must preserve colimits.

At the left end of the presheaf category chain, $\Lambda$ does not preserve the terminal object. The terminal categorical presheaf has singleton value everywhere, whereas $(\Lambda 1)(S)=\varnothing$ because $S\ne \varnothing$. Thus $\Lambda$ cannot itself be a right adjoint. At the right end, $\nabla$ does not preserve the initial object: $(\nabla \varnothing )(\varnothing )=\{\ast \}$ since the empty open set is proper. Thus $\nabla$ cannot itself be a left adjoint.

For the category of small categories chain, take two functors from the terminal category $1$ to the two-object indiscrete category $I\{0,1\}$, selecting its distinct objects. Their equalizer is the empty category. But their images under $C={\pi }_0$ are the same function between singleton sets, whose equalizer is a singleton. Hence $C$ does not preserve equalizers and has no further left adjoint. Finally, $I1⨿I1$ is the two-object discrete category, while $I(1⨿1)$ has arrows between its distinct objects. The canonical comparison is not an isomorphism of categories, so $I$ fails to preserve this coproduct and has no further right adjoint.