nLab
fiber product

Definition

A fibre product or fiber product is a product in a slice category $𝒞\downarrow C$. The fibre product of two morphisms $f:A\to C$, $g:B\to C$ is the same as their pullback; accordingly, a fiber product of more than two morphisms $f_i:A_i\to C$ is often called a wide pullback.

More explicitly, for $f:A\to C$ and $g:B\to C$ two morphisms in a category $𝒞$, the fiber product $A{\times }_{C}B$ of $A$ with $B$ over $C$ is, if it exists, the pullback

$$\begin{array}{ccc}A{\times }_{C}B& \to & B\\ \downarrow & & {\downarrow }^g\\ A& \stackrel{f}{\to }& C\end{array}.$$\array{ A \times_C B &\to& B \\ \downarrow && \downarrow^g \\ A &\stackrel{f}{\to}& C } \,.

This term comes from thinking of $A$ and $B$ as bundles over $C$; then the fiber of $A{\times }_{C}B$ over a generalized element $x$ of $C$ is the product of the fibers of $A$ and $B$ over $x$. In other words, the fiber product is the product taken fiber-wise.

Of course, the fiber of $A$ at the generalized element $x:I\to C$ is itself a fibre product $I{\times }_{C}A$; the terminology depends on your point of view.

Examples

In $𝒞=$ Set, the fiber product is given by the usual formula

$$A{\times }_{C}B=\{(a,b)\in A\times B\mid f(a)=g(b)\}.$$A \times_C B = \left\{ (a,b) \in A \times B \;|\; f(a) = g(b) \right\} \,.

Related concepts

• pullback, base change

• lax pullback, comma object

• (∞,1)-pullback, homotopy pullback