2
$\begingroup$

I was wondering if the product category makes sense for an arbitrary family of (non-necessary small) categories. It is clear that given any finite number of categories, you can construct their product category, even if some of the categories are not small. Furthermore, if $\{\mathcal C_i\}_{i\in I}$ is a family of small categories indexed by a set $I$, you should be able to construct their product $\prod_{i\in I}\mathcal C_i $ which, if I am not wrong, is a subcategory of the functor category $\mathcal C^I$, where $\mathcal C=\bigcup_{i\in I} \mathcal C_i$ (I am not an expert in category theory but according to Wiki $\mathbf{Cat}$ is a Cartesian closed category which means that the exponential of any two objects always exists).

So, my question is: what happen if some of the categories $\mathcal C_i$ is not small (for example if it is $\mathbf{Set}$)? Can we still define the product category of this family?

Remark. As I have already said I'm not an expert in category theory. I'm studying only the basics of the theory as a tool for understanding products, pullbacks and so on. So, As Mac Lane says in his book, my category theory is subordinate to set theory (where I am not an expert either). With this I mean I don't know anything about foundations, so that you should avoid an answer based on this (if my question was related with).

Thanks for your help.

$\endgroup$
1
$\begingroup$

$I$ is a set so there is no problem. You can define in a natural way the category product and your only problem is to verify that

$hom((A_i)_{i\in I} ,(B_j)_{j\in I})=$

$=\{(f_i)_{i\in I}: f_i\in hom(A_i,B_i)\}$

is a set, that is true because $I$ and $hom(A_i,B_i)$ are sets for every $i\in I$

$\endgroup$
  • $\begingroup$ First of all, thank you very much for your anwer. I have only one question. When all the categories are small I can consider the union of all of them which is small and then as I have said the functor category exists. But if some of the categories are not small, then the union is not a small category. How can I guarantee the existence of the product (if I can) in this case? Thanks $\endgroup$ – Dog_69 Jul 3 '18 at 11:20
  • 1
    $\begingroup$ The functor category $D^C$ always exists for a small $C$: the size of $D$ does not matter. If you want an explanation for that, it will most likely depend on the foundations you choose; but you said you wanted to avoid those (for instance $Set^C$ is always a category if $C$ is small) $\endgroup$ – Max Jul 3 '18 at 11:32
  • $\begingroup$ @Max: Perfect. It's enough Thanks. $\endgroup$ – Dog_69 Jul 3 '18 at 11:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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