# Category Theory: Does the existence of $X\times (Y\times Z)$ imply the existence of $X\times Y$? [closed]

Let $$X\times Y$$ be the product of objects $$X$$ and $$Y$$ in a category $$C$$, defined by the universal property of products. Is it true that the existence of the product $$X\times (Y\times Z)$$ implies the existence of $$X\times Y$$?

No. Consider the following minimal example. Consider the category with objects

$$A, X, Y, Z, Y \times Z, X \times (Y \times Z)$$

and the morphisms

\begin{align} \pi_1: X \times (Y \times Z) &\rightarrow X \\ \pi_2: X \times (Y \times Z) &\rightarrow Y \times X \\ \pi'_1: Y \times Z &\rightarrow Y \\ \pi'_2: Y \times Z &\rightarrow Z \\ a_1:A &\rightarrow X \\ a_2: A &\rightarrow Y \end{align}

and their compositions i.e. $$\pi'_1 \circ \pi_2$$ and $$\pi'_2 \circ \pi_2$$.

• In this example, $X\times (Y\times Z)$ is the product of $X$ and $Y$. But you can fix it by adding another object with morphisms to both $X$ and $Y$. – Jeremy Rickard Mar 20 at 9:09
• Jeremy Rickard, that's what I just thought of... – The Giraffe Guy Mar 20 at 9:13
• Completely right. Thx. Edited. – G. Chiusole Mar 20 at 9:15

No. Consider the partial order $$\mathbb{N}\cup\{\infty_1, \infty_2\}$$ where $$\infty_1$$ and $$\infty_2$$ are incomparable to each other and greater than all natural numbers. Taken to be a category in the usual way (a unique morphism $$x \to y$$ if $$x \le y$$), products here are greatest lower bounds.

Then if $$X = \infty_1$$ and $$Y = \infty_2$$, the greatest lower bound of $$X$$ and $$Y$$ doesn't exist (the lower bounds are all natural numbers). However, if we take $$Z = 0$$, then $$Y \times Z$$ exists and equals $$0$$. Similarly, $$X \times (Y \times Z)$$ exists and equals $$0$$.