Can You Have a Category of Categories Without Encountering Russell's Paradox?

[Diffy]

Can you have a Category where the objects are Categories and the mappings are functors?

If you can, then can one have the Category of all categories, or do you run into some form of Russell's Paradox?

 

[Hurkyl]

do you run into some form of Russell's Paradox?

I'm pretty sure you do. The usual procedure is to invoke some 'size' tricks similar to what happens in set theory or formal logic. If we do foundations set theoretically...

We assume the existence of a large cardinal number bigger than N. In other words, we refine the normal hierarchy of 'size':

empty -- finite -- infinite​

to become

empty -- finite -- small -- large​

(Okay, that's not quite right. "infinite" should also include the finite sets)

Now, you let Set denote the category of all small sets (note that Set satisfies all of the ordinary axioms of set theory), and you let Cat be the category of all small categories. Both Set and Cat are large, so you don't have any Russell's paradox issues.

Now, if you want to reason about large categories, you might then consider things like the category of all large sets, and the category of all large categories. Both of these categories are proper classes, so again we don't have any Russell's paradox issues. (CWM calls these 'metacategories')

If you like, you can iterate this idea -- if you assume two large cardinals, you can get a hierarchy

empty -- finite -- small -- large -- 'superlarge'​

and then the category of all large categories is a superlarge category, and we can consider things like the metacategory of superlarge categories.

Normally, you don't bother iterating much unless you get to higher category theory. (CWM invokes only one large cardinal)