What does the notations $X \times_{Z} Y$ mean?

Question

Right now I am learning how to create commutative diagrams in Latex, and since I am also studying category theory right now, I have come across the notation before in category theory texts like Categories For The Working Mathematician. Here, I am referencing the following: http://ctan.math.washington.edu/tex-archive/graphics/pgf/contrib/tikz-cd/tikz-cd-doc.pdf. On the third page, first diagram, you have $X \times_{Z} Y$. Can someone give an explanation on the notation?

Answer 1

This is the fibered product, or pullback, of $X$ and $Y$.

It is the limit of a diagram consisting of two arrows $f: X \to Z$ and $g: Y \to Z$.

Answer 2

If you have objects $X,Y,Z$ and morphisms $X\overset{f}{\to}Z,Y\overset{g}{\to}Z$, then the fibered product $$X\times_{f,Z,g}Y$$ (or $X\times_ZY$ if $f,g$ are understood) is an object coming with two morphisms $X\times_ZY\overset{p_1}{\to}X,X\times_ZY\overset{p_2}{\to}X$, with $f\circ p_1=g\circ p_2$, checking the universal property of your diagram. Thus, for every object $T$ with morphisms $T\overset{q_1}{\to}X, T\overset{q_2}{\to}Y$ with $q_1\circ f=q_2\circ g$, you will have a unique morphism $T \overset{h}{\to}X\times_ZY$ such that $q_1=p_1\circ h$ and $q_2=p_2\circ h$.

If you want to manipulate the concept a little bit, then for example the category Set, where objects are sets and morphisms are applications between sets, has a fibered product. One realisation can be explicitely given by

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

Still in Set, you can check that $X\times_{\{\ast\}} Y$ is isomorphic to $X\times Y$, where $\{\ast\}$ is a set with one element.