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The wiki page on complete category states:

The existence of all limits (even when J is a proper class) is too strong to be practically relevant. Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other.

Why is that? Can we interpret any object in such a category as a limit candidate of some large diagram?

(Also, isn’t this little fact a nice motivation to really care about the distinction between sets and classes in category theory? Often enough I encountered the attitude “Yeah, we don’t care about classes or sets, just think of all those things as collections” – should I discard that attitude?)

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2 Answers 2

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Let $C$ be a category with all (large) limits. Assume that there are two morphisms $f,g : a \to b$, $f \neq g$. If $J$ is a class (or large set), then there are at least $2^J$ morphisms $a \to b^J$, where $b^J := \prod_{j \in J} b$. Applying this to $J:=\mathsf{Mor}(C)$, we obtain an injection $2^J \hookrightarrow J$, which is a contradiction.

And yes, this is one of the reasons why I think that it is important to distinguish between sets and classes (or large sets). In category theory it is often convenient to work with universes, sometimes even more of them contained in each other. This makes it possible to work with arbitrary "levels" of largeness.

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  • $\begingroup$ As always: Thank you, Martin. : D $\endgroup$
    – k.stm
    Nov 29, 2014 at 13:45
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I do not have enough reputation to comment, but I just wanted to say that this fact is due to Freyd. You might be interested in the following little paper as well.

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