The Stacks project

4.7 Examples of fibre products

In this section we list examples of fibre products and we describe them.

As a really trivial first example we observe that the category of sets has fibre products and hence every morphism is representable. Namely, if $f : X \to Y$ and $g : Z \to Y$ are maps of sets then we define $X \times _ Y Z$ as the subset of $X \times Z$ consisting of pairs $(x, z)$ such that $f(x) = g(z)$. The morphisms $p : X \times _ Y Z \to X$ and $q : X \times _ Y Z \to Z$ are the projection maps $(x, z) \mapsto x$, and $(x, z) \mapsto z$. Finally, if $\alpha : W \to X$ and $\beta : W \to Z$ are morphisms such that $f \circ \alpha = g \circ \beta $ then the map $W \to X \times Z$, $w\mapsto (\alpha (w), \beta (w))$ obviously ends up in $X \times _ Y Z$ as desired.

In many categories whose objects are sets endowed with certain types of algebraic structures the fibre product of the underlying sets also provides the fibre product in the category. For example, suppose that $X$, $Y$ and $Z$ above are groups and that $f$, $g$ are homomorphisms of groups. Then the set-theoretic fibre product $X \times _ Y Z$ inherits the structure of a group, simply by defining the product of two pairs by the formula $(x, z) \cdot (x', z') = (xx', zz')$. Here we list those categories for which a similar reasoning works.

1. The category $\textit{Groups}$ of groups.

2. The category $G\textit{-Sets}$ of sets endowed with a left $G$-action for some fixed group $G$.

3. The category of rings.

4. The category of $R$-modules given a ring $R$.