Let $X, Y, Z$ be affine varieties defined over $k$ an algebraically closed field. I was wondering how we know that $$ (X \times_k Y) \times_k Z = X \times_k (Y \times_k Z)? $$
Actually I'm not even sure if it's supposed to be "=" here.. What should it be? Any explanation for the two parts would be appreciated. Thank you.
Fiber products in any category are naturally associative.
Let $X,Y,Z,A,B$ be objects in a category $\mathcal{C}$ with maps $f:X\to A$, $g:Y\to A$, $h:Y\to B$, $j:Z\to B$.
Then the fiber products $(X\times_A Y)\times_B Z$ and $X\times_A(Y\times_B Z)$ both represent the functor from $\mathcal{C}^{\text{op}}$ to $\mathbf{Set}$ which takes an object $W$ to the collection of all triples of morphisms $(\alpha,\beta,\gamma)$, where $\alpha: W\to X$, $\beta: W\to Y$, $\gamma:W\to Z$ such that $f\circ \alpha = g\circ \beta$ and $h\circ \beta = j \circ \gamma$. Since they both represent the same functor, they are naturally isomorphic.