Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 2

A categorical limit of $D:\mathcal{J}\to \mathcal{C}$ is a terminal cone $(L,(p_j))$: every cone $(X,(x_j))$ has a unique map $X\to L$ commuting with all legs.

Assume all small products and equalizers exist. PutDefine $s,t:P\rightrightarrows Q$ so that the $u:i\to j$ components areA map $X\to P$ equalizes $s,t$ exactly when its components form a cone over $D$. Therefore $\mathrm{eq}(s,t)$ represents cones and is the limit. This is the construction of small limits from products and equalizers.

Let $D:\mathcal{J}\to \mathbf{Fld}$ be connected and nonempty. The limit of the underlying diagram of commutative rings is the subringIf $x\in L$ is nonzero in one component, it is nonzero in every component: field homomorphisms are injective, and connectedness propagates this fact along zigzags. Hence the componentwise inverses $(x_j^{-1})$ are defined and compatible. Thus $L$ is a field. Since the inclusion $\mathbf{Fld}\hookrightarrow \mathbf{CRng}$ is full, the same cone is limiting in the category of fields.

If $\mathcal{J}$ is disconnected, choose two components and use the constant field ${\mathbb{F}}_2$ on one and ${\mathbb{F}}_3$ on the other. There is no cone in fields because its apex would map to fields of two different characteristics. Hence $\mathbf{Fld}$ does not have all limits of any disconnected shape.

Consider one connected component $K$ of the category of cocones under a small diagram $D:\mathcal{J}\to \mathbf{Fld}$. Replace every apex by the subfield generated by the images of the fields $D(j)$. These generated fields have cardinality bounded in terms of the small diagram, so they admit a small skeleton, cofinal within $K$.

The apex functor on this small connected category has a limit $L_K$ in $\mathbf{Fld}$ by part (b). For each $j$, the maps from $D(j)$ to all apexes form a compatible cone and therefore induce $D(j)\to L_K$. These maps form a cocone ${\lambda }_K$ under $D$. Its limiting projections give a unique morphism from ${\lambda }_K$ to every cocone in $K$, so ${\lambda }_K$ is initial in that component. Choosing one ${\lambda }_K$ for each component gives a multicolimit.