definition of a fiber product

DEFINITION 6.1   Let $\mathcal{C}$ be a category. Let $X,Y,Z$ be objects of $\mathcal{C}$ . Assume that morphisms $f: X\to Z$ and $g:Y\to Z$ are given. Then the fiber product $X\times_Z Y$ (more precisely,

$$X\times_{f,Z,g}Y$$

) is defined as an object $W$ together with morphisms

$$p:W \to X, \quad q: W\to Y$$

such that

$$p \circ f= q\circ g$$

which is universal in the following sense.

For any $W_1 \in \operatorname{Ob}(\mathcal{C})$ together with morphisms

$$p_1:W_1 \to X, \quad q_1: W_1\to Y$$

such that

$$p_1 \circ f= q_1\circ g,$$

there exists a unique morphism $h: W_1\to W$ such that

$$p\circ h=p_1 , q\circ h=q_1$$

holds.

Using the usual universality argument we may easily see that the fiber product is, if exists, unique up to a unique isomorphism.

EXAMPLE 6.2   Fiber products always exists in the category $(\operatorname{TOP})$ of topological spaces. Namely, let $X,Y,Z \in \operatorname{Ob}(\operatorname{TOP})$ . Assume that morphisms(=continuous maps) $f: X\to Z$ and $g:Y\to Z$ are given. Then we consider the following subset $S$ of $X\times Y$ .

$$S= (f,g)^{-1} (\Delta_Z) (\subset (X\times Y))$$

$$= \{(x,y)\in X\times Y; f(x)=g(y)\}.$$

We equip the set $S$ with the relative topology. Then $S$ plays the role of the fiber product $X\times_Z Y$ . (The morphisms $p,q$ being the (restriction of) projections.)