Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 25 / 2

Let $D:J\to \mathcal{C}$ be a finite diagram in a category. Form the finite products in a categoryDefine $s,t:P\rightrightarrows Q$ by ${\pi }_{a}s=D(a){\pi }_j$ and ${\pi }_{a}t={\pi }_k$. Take their equalizer $e:L\to P$. The maps ${\lambda }_j={\pi }_{j}e$ satisfy $D(a){\lambda }_j={\lambda }_k$, so they constitute a categorical cone.

Any other categorical cone $x_j:X\to D(j)$ gives a unique $x:X\to P$ by the product in a category property. Its cone equations say $sx=tx$, so the equalizer gives a unique $\bar{x}:X\to L$ with $e\bar{x}=x$. The projections then give ${\lambda }_j\bar{x}=x_j$, and uniqueness follows from both universal properties. Thus this equalizer of two product maps is the finite limit of the diagram in a category.

The empty indexing category is included: its empty product in a category is a terminal object, which is the empty categorical limit. Hence no nonemptiness restriction has entered the construction.

With a terminal object $1$, the pullback in a category of $A\to 1\leftarrow B$ is the binary product in a category $A\times B$. Repetition gives all nonempty finite products in a category, and $1$ supplies the empty one.

For $f,g:A\rightrightarrows B$, form the diagonal ${\Delta }_B:B\to B\times B$ and pull it back along $⟨f,g⟩:A\to B\times B$. Write the resulting maps as $e:E\to A$ and $h:E\to B$. The equation $⟨f,g⟩e={\Delta }_{B}h$ says $fe=ge=h$. A map $x:X\to A$ with $fx=gx$ pairs with $fx:X\to B$ to give a unique map into this pullback in a category. Thus $e$ is the equalizer of $f,g$.

We now have the finite products in a category and equalizers needed in part (a). A terminal object and pullbacks in a category therefore suffice for all finite limits.

An object of the slice category $\mathcal{C}/A$ is a morphism $x:X\to A$; a map from $x$ to $y:Y\to A$ is $u:X\to Y$ with $yu=x$. The identity morphism $1_A:A\to A$ is a terminal object of this slice category, even when $\mathcal{C}$ itself has no terminal object.

For a cospan $X\stackrel{u}{\to }Z\stackrel{v}{\leftarrow }Y$ in the slice category, form its underlying pullback in a category $P$ in $\mathcal{C}$, with maps $p:P\to X$ and $q:P\to Y$. If $x,y,z$ are the structure maps to $A$, then $xp=zup=zvq=yq$. This common map equips $P$ with its slice structure. An underlying mediating map into $P$ is automatically over $A$, because its composite with $xp$ is the prescribed structure map. Hence the same square is a pullback in a category in $\mathcal{C}/A$.

Part (b) applies to the slice's terminal object and pullbacks in a category. Every slice category has all finite limits when the ambient category has pullbacks in a category. The finite completeness of slice categories result needs no ambient terminal-object hypothesis.