The Stacks project

4.8 Fibre products and representability

In this section we work out fibre products in the category of contravariant functors from a category to the category of sets. This will later be superseded during the discussion of sites, presheaves, sheaves. Of some interest is the notion of a “representable morphism” between such functors.

Lemma 4.8.1. Let $\mathcal{C}$ be a category. Let $F, G, H : \mathcal{C}^{opp} \to \textit{Sets}$ be functors. Let $a : F \to G$ and $b : H \to G$ be transformations of functors. Then the fibre product $F \times _{a, G, b} H$ in the category $\textit{PSh}(\mathcal{C})$ exists and is given by the formula

\[ (F \times _{a, G, b} H)(X) = F(X) \times _{a_ X, G(X), b_ X} H(X) \]

for any object $X$ of $\mathcal{C}$.

Proof. Omitted. $\square$

As a special case suppose we have a morphism $a : F \to G$, an object $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and an element $\xi \in G(U)$. According to the Yoneda Lemma 4.3.5 this gives a transformation $\xi : h_ U \to G$. The fibre product in this case is described by the rule

\[ (h_ U \times _{\xi , G, a} F)(X) = \{ (f, \xi ') \mid f : X \to U, \ \xi ' \in F(X), \ G(f)(\xi ) = a_ X(\xi ')\} \]

If $F$, $G$ are also representable, then this is the functor representing the fibre product, if it exists, see Section 4.6. The analogy with Definition 4.6.4 prompts us to define a notion of representable transformations.

Definition 4.8.2. Let $\mathcal{C}$ be a category. Let $F, G : \mathcal{C}^{opp} \to \textit{Sets}$ be functors. We say a morphism $a : F \to G$ is representable, or that $F$ is relatively representable over $G$, if for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and any $\xi \in G(U)$ the functor $h_ U \times _ G F$ is representable.

Lemma 4.8.3. Let $\mathcal{C}$ be a category. Let $a : F \to G$ be a morphism of contravariant functors from $\mathcal{C}$ to $\textit{Sets}$. If $a$ is representable, and $G$ is a representable functor, then $F$ is representable.

Proof. Omitted. $\square$

Lemma 4.8.4. Let $\mathcal{C}$ be a category. Let $F : \mathcal{C}^{opp} \to \textit{Sets}$ be a functor. Assume $\mathcal{C}$ has products of pairs of objects and fibre products. The following are equivalent:

  1. the diagonal $\Delta : F \to F \times F$ is representable,

  2. for every $U$ in $\mathcal{C}$, and any $\xi \in F(U)$ the map $\xi : h_ U \to F$ is representable,

  3. for every pair $U, V$ in $\mathcal{C}$ and any $\xi \in F(U)$, $\xi ' \in F(V)$ the fibre product $h_ U \times _{\xi , F, \xi '} h_ V$ is representable.

Proof. We will continue to use the Yoneda lemma to identify $F(U)$ with transformations $h_ U \to F$ of functors.

Equivalence of (2) and (3). Let $U, \xi , V, \xi '$ be as in (3). Both (2) and (3) tell us exactly that $h_ U \times _{\xi , F, \xi '} h_ V$ is representable; the only difference is that the statement (3) is symmetric in $U$ and $V$ whereas (2) is not.

Assume condition (1). Let $U, \xi , V, \xi '$ be as in (3). Note that $h_ U \times h_ V = h_{U \times V}$ is representable. Denote $\eta : h_{U \times V} \to F \times F$ the map corresponding to the product $\xi \times \xi ' : h_ U \times h_ V \to F \times F$. Then the fibre product $F \times _{\Delta , F \times F, \eta } h_{U \times V}$ is representable by assumption. This means there exist $W \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, morphisms $W \to U$, $W \to V$ and $h_ W \to F$ such that

\[ \xymatrix{ h_ W \ar[d] \ar[r] & h_ U \times h_ V \ar[d]^{\xi \times \xi '} \\ F \ar[r] & F \times F } \]

is cartesian. Using the explicit description of fibre products in Lemma 4.8.1 the reader sees that this implies that $h_ W = h_ U \times _{\xi , F, \xi '} h_ V$ as desired.

Assume the equivalent conditions (2) and (3). Let $U$ be an object of $\mathcal{C}$ and let $(\xi , \xi ') \in (F \times F)(U)$. By (3) the fibre product $h_ U \times _{\xi , F, \xi '} h_ U$ is representable. Choose an object $W$ and an isomorphism $h_ W \to h_ U \times _{\xi , F, \xi '} h_ U$. The two projections $\text{pr}_ i : h_ U \times _{\xi , F, \xi '} h_ U \to h_ U$ correspond to morphisms $p_ i : W \to U$ by Yoneda. Consider $W' = W \times _{(p_1, p_2), U \times U} U$. It is formal to show that $W'$ represents $F \times _{\Delta , F \times F} h_ U$ because

\[ h_{W'} = h_ W \times _{h_ U \times h_ U} h_ U = (h_ U \times _{\xi , F, \xi '} h_ U) \times _{h_ U \times h_ U} h_ U = F \times _{F \times F} h_ U. \]

Thus $\Delta $ is representable and this finishes the proof. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0021. Beware of the difference between the letter 'O' and the digit '0'.