Qiaochu: I haven't thought hard about it. Everyone: it's not that the existence of such things is particularly in question (I fully believe it). But all the examples I've been provided with are handwavy. What I want is a very explicit example, and hopefully a reference.

Thanks, Richard; that seems very reasonable. I'll see whether I can get hold of that reference.

That's very interesting Mark (and thank you), but an explicit example would be very nice. Explicit geometric examples would be most welcome!

My thanks to those who answered Jim Stasheff's question. It arose in a discussion over at the nLab (and of course we're interested in the generalization to star-shaped open domains, addressed by Dan Ramras and Robin Chapman below).

You're welcome, Pete -- I was puzzled too, not knowing what Rota could have meant by "necessary and sufficient conditions" for CRT, since as you had pointed out, the more familiar version of CRT is a very simple and general thing. So I googled it, and this was one of the results. But I'm still puzzled as to why Rota seems to think that this result is something Grothendieck would have especially benefited from, or whether there's more to the story that he's not telling us.

You're right, Sadeq, that I didn't notice that condition. However, the statement is still incorrect if $a, b > 1$ are arbitrary numbers (take $a = 5$ and $b = 1.001$). Or, better yet, take $a = 4$, $b = 2$.

I don't know if my suggestion would actually "work"; I suspect it would work only if the child rather enjoys mathematics. But it could be fun to talk about logarithmic spirals (they are beautiful and occur in nature). I also remember some nice related visuals from Donald Duck in MathMagic Land.

I was not aware of that, J.M. -- thanks!

Sadeq, your penultimate statement is wrong: consider $a = 5$ and $b = 1$. In your final sentence, you either need to replace $>$ by $\geq$ or specify that $x \neq e$.

Thanks for a fresh and flavorful answer, Andrew! And one that helps restore dignity to those who choose category theory as a focus of research.