Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I've (partially) read some books about category theory. But only now attempting to put it into research practice I noticed that I do not really understand direct products.

Consider a product $A\times B$ of arrows $x_1: X\rightarrow A$ and $x_2: X\rightarrow B$.

Let now the category Set and $A=\varnothing$. Then there are no arrow $x_1: X\rightarrow A$.

So direct product (in Set) with an empty set does not exist. (I previously though that it exist. Was I wrong?)

I understand something in a wrong way. Please help to understand it properly.

share|improve this question
    
I assume you're writing $X$ for $A \times B$. The conclusion has to be that $\emptyset \times B = \emptyset$. Check that this satisfies the universal property! –  Dylan Moreland Jul 22 '12 at 19:01
    
@DylanMoreland: No, $X$ is an arbitrary set. –  porton Jul 22 '12 at 19:03
4  
There is an arrow $X \to \emptyset$ if $X = \emptyset$ ! –  Ted Jul 22 '12 at 19:04
    
@porton I don't see how that shows that $A \times B$ does not exist. The definition of the product of two elements in a category does not say that for any $X$ in the category there exists an arrow $X \to A$. And with $A = \emptyset$ there is only one such $X$ and one such arrow. –  Dylan Moreland Jul 22 '12 at 19:05

2 Answers 2

To check that $\emptyset$ satisfies the categorical definition of the product $M\times\emptyset$, we need to specify morphisms $\pi_1:\emptyset\to M$ and $\pi_2:\emptyset\to\emptyset$ and then check the universal property. The morphisms are easy, since there is only a single map from $\emptyset$ to any set, namely the empty map. Now given any set $Y$ and morphisms $f_1:Y\to M$ and $f_2:Y\to\emptyset$, there must be a unique $f:Y\to\emptyset$ such that $\pi_1\circ f=f_1$ and $\pi_2\circ f=f_2$. Note that $Y$ must be empty, since otherwise there are no such morphisms $f_2$. Since $Y$ is empty, $f_1$ and $f_2$ are empty maps, and there is indeed a unique $f:Y\to\emptyset$ with the required property, namely the empty map. Thus the universal property is fulfilled, and $\emptyset$ is indeed the categorical product $M\times\emptyset$.

share|improve this answer

I read this as saying that the direct product with an empty set is empty. This is different than saying the direct product does not exist.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.