Generalizations of fibre products

Question

For maps $f:X\to Z$ and $g:Y\to Z$ of topological spaces, we can define fibre product as $X\times_{Z}Y=\{(x,y)\in X\times Y \mid f(x)=g(y)\}$. I was wondering if there is a generalization of this concept. More precisely, if we have $f_i:X_i\to Z$ for $1\leq i\leq n$, then can we define generalized fibre product as a topological space $S=\{(x_1,\dots,x_n)\in \prod_{i=1}^nX_i \mid f_i(x_k)=f_j(x_l) ,i\neq j ~\text{and}~ k\neq l \}$?

Answer 1

Obviously the space you wrote down exists, it's just a (possibly empty) subspace of a finite product, so in one sense the answer to "can we define [...]" is trivially yes.

But of course this begs the question of "in what ways is this a generalization $X \times_Z Y$", and this in turn begs the question of what properties of the fibre product we should consider in the first place. Here's one suggestion: The fibre product $X \times_Z Y$ satisfies a universal property: Given a space $W$ and maps $\alpha\colon W \to X$, $\beta\colon W \to Y$ such that $f \circ \alpha = g \circ \beta$, then there exists a unique map $\gamma\colon W \to X \times_Z Y$ such that $\alpha = \pi_X \circ \gamma$ and $\beta = \pi_Y \circ \gamma$ (where $\pi_X\colon X \times_Z Y \to X$ is the projection, and likewise for $\pi_Y$).

The space $S$ also has a property like this: Given $W$ and maps $\alpha_i\colon W \to X_i$ such that $f_i \circ \alpha_i = f_j \circ \alpha_j$ for all $i, j$, there exists a unique map $\gamma\colon W \to S$ such that $\alpha_i = \pi_{X_i} \circ \gamma$. (There's an obvious choice for this map, namely $\gamma(w) = (\alpha_1(w), \ldots, \alpha_n(w))$. I leave the proof that this has the stated universal property as an exercise.) Thus, $S$ and $X \times_Z Y$ are in a (pretty strong!) sense alike.

What we've stumbled upon here is the categorial notion of limit: $X \times_Z Y$ is the limit of the diagram defined by the morphisms $f$ and $g$ (this particular limit is also often called a pullback), whereas $S$ is the limit of the diagram given by the $f_i$ (strictly speaking the projection maps $\pi_{X}$, $\pi_{Y}$, and $\pi_{X_i}$, respectively, are also part of the limit data, but here they are clear). I refer you to, say, Emily Riehl's Category Theory in Context for a general exposition (or any other basic text on category theory of your liking).