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

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?

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 $$g: Y \to Z$$.

• OP might want to know where the $Z$ is. – Randall May 22 at 15:17
• I'm not sure what you mean by that. – Lukas Kofler May 22 at 15:18
• Your answer contains no mention of "$Z$" unless I click on the link, which could be altered in the future so as to make no sense. – Randall May 22 at 15:19
• Right. I've expanded my answer. – Lukas Kofler May 22 at 15:20

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.

• The notation $p_X$ and $p_Y$ is not a good choice, since if $X=Y$, then we will have the same symbol for two different things. – Nex May 22 at 19:30
• Indeed! Thanks for pointing that out. – Balloon May 22 at 19:32