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Mac Lane works in CWM with a universe $U$. It has certain properties, and one of them is that $U$ is transitive. All sets belonging to $U$ are small sets and subsets of $U$ are classes. Classes that are not small sets are proper classes. Now I quote (from pg 22):


'...a category is small if the set of its arrows and the set of its objects are both small sets;'


Secondly I quote (from pg 23)


'A large category is one in which both the set of objects and the set of arrows are classes (proper or otherwise).'


The word 'otherwise' here gives me the impression that every small category is also a large category. The transitivity of $U$ implies that small sets - as elements of $U$ - are also subsets of $U$, i.e. classes (that are not proper). Is my interpretation correct? If so then I wonder: are there any categories that are not large? If the answer is 'no' then what is the use of giving every category the label 'large'?

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You're thoughts are correct. He probably calls them "large" just to emphasize, that they are maybe not small. In many situations in category theory is it fundamental, whether the category you're working with is small oder not and many things won't work with categories, that are not small. So "large" means intuitively "be careful". –  archipelago Sep 15 '13 at 20:31
    
I think better terminology would be "potentially large" –  goblin Sep 15 '13 at 20:58
    
Are finite sets countable? Sometimes they are, sometimes they aren't... –  Zhen Lin Sep 15 '13 at 22:58

1 Answer 1

up vote 2 down vote accepted

Probably Mac Lane meant large category for not necessarily small, meaning that small category are large categories too. So I think your interpretation is correct.

Anyway based on this very reliable source it seems that's not universally accepted if by large category one should mean a category which is not small or a category which could (but does't have to) be small.

About the necessity of the term large category, the same Mac Lane explain that there are some categories which aren't large: for instance the category of classes and function between them. This isn't a large category because the collection of the objects isn't a class (neither a proper one), for the Cantor's theorem for classes.

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