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

Suppose $X\cong \mathcal{C}(-,R)$. As a functor on the opposite category, a representable presheaf preserves every existing categorical limit: a colimit in $\mathcal{C}$ is defined by the bijection between morphisms from its vertex into $R$ and compatible families of morphisms from its diagram objects into $R$. This remains valid for a possibly large diagram in a category whenever that colimit exists and the compatible-family collection is the corresponding set.

The category of elements has a terminal object $(R,r)$, where $r$ is the image of $1_R$ under the representation. For each $(C,x)$ let $u_{C,x}:C\to R$ be its unique morphism to $(R,r)$. These form a cocone for the forgetful functor $G$. Any competing cocone $v_{C,x}:C\to A$ satisfiesThe component at $(R,r)$ uniquely determines the mediating morphism. Thus the representing object is the colimit of the elements projection:

Let $u_{C,x}:C\to R$ be a colimit cocone for $G:\mathrm{Elts}(X)\to \mathcal{C}$. Here the preservation hypothesis must include the categorical limit of $G^{\mathrm{op}}$ in ${\mathcal{C}}^{\mathrm{op}}$; this indexing category can be large. For a morphism $f:(C,x)\to (D,y)$ of the category of elements, the defining equation $X(f)(y)=x$ says that the family of distinguished elements $(x{)}_{(C,x)}$ is compatible. Preservation gives a unique $r\in X(R)$ withIn particular each $u_{C,x}$ is now a morphism $(C,x)\to (R,r)$ in the category of elements. Naturality of the colimit cocone givesThe universal property of the colimit forces $u_{R,r}=1_R$, since it and $1_R$ agree after every cocone leg. If $f:C\to R$ also satisfies $X(f)(r)=x$, it is a morphism $(C,x)\to (R,r)$, and cocone compatibility now yields $u_{C,x}=u_{R,r}f=f$. Thus $(R,r)$ is a universal element. By the preceding representability criterion, the presheaf is represented by the colimit object:The size qualification matters. If preservation means only small categorical limits, the implication as printed is false for general locally small categories. For an explicit counterexample, let $\mathcal{C}$ be the ordered category of all ordinals with an extra greatest object $\infty$. Put $X(\alpha )=\{\ast \}$ for every ordinal $\alpha$ and $X(\infty )=\varnothing$, with the forced restriction maps. Every small colimit in $\mathcal{C}$ is the supremum of its object values: it is an ordinal unless the diagram contains $\infty$. Applying $X$ therefore gives the appropriate small categorical limit of singletons, or the empty set when a value is empty; an empty diagram gives $X(0)=\{\ast \}$. Hence $X$ preserves all small categorical limits.

Its category of elements consists of all ordinals, whose projection has the large colimit $\infty$ in $\mathcal{C}$. Nevertheless, no ordinal represents $X$, because its representable vanishes on larger ordinals; $\infty$ does not represent it either, since $\mathcal{C}(\infty ,\infty )$ is nonempty while $X(\infty )$ is empty. The valid proof consequently uses preservation of the displayed possibly large limit, or a smallness hypothesis making that diagram small.