@Shulman: I wouldn't expect a different foundation to play exactly the same role as the original, but one would expect there to be a great deal of commonality. Does this mean that ETCS is entirely standalone? So although its inspired by category theory that scaffolding can be taken away. I find this point a little confusing: why do this? Isn't there an 'elementary theory of categories' that doesn't rely on ZFC (I thought Maclane tries to do this in his book) then why not keep to the natural order of 'inspiration'? Unless of course there is no such elementary theory as it runs into difficulties

I just reread this, and realised this is the kind of answer that I was looking for.

Galois categories inspired the Tannakian categories formalism that reconstructs an affine group scheme from its finite-dimensional representations.

@Joel:Thanks for clarifying Fukuyamas title. I thought Fukuyama was also stating if not explicitly, then implicitly that liberal democracies were the endpoint of the evolution of political forms of a state? Quite, except when the "I" is an influential person in the field, remarks such as these are more influential and drive people away.

@Humphreys: Thanks for the online reference. I do realise that a universal construction is only for characterisation, and automatically proves uniqueness, so long as existence is shown. I have online access, but not a useful library access, unfortunately.

@Shanmukha_Srinivasan: My silence has more to with getting a job than anything else:). It would probably have been better to have waited until my circumstances were a bit more stable. The question popped into my head sometime after I first learnt about Lie Groups/Algebras, and I ran across Verma Module on Wikipedia, and that in itself is a while ago. (I used Dragon Milicic online notes which I found very useful). I agree with Tom Leinsters comments too.

I entirely agree with you, and I think Theo makes a very good point. I don't think I've ever seen a concrete tensor...

@Ryan: I'd agree it isn't focused enough, but I'd dispute its unmotivated.

@user49437: my first question was reacting against Michors assertion that Banach Manifolds aren't interesting. Obviously they're not flexible enough notion for the purposes he wants to put them to.

@Evans: you mean infinite-dimensional vector spaces?

@Geschke: couldn't this question be closed off as 'rough duplicate', rather than off-topic.

@Leinster: Thanks for the suggestions; I'll amend appropriately. (Had it been not so far into the small hours, I would have done so when I wrote it.)

@Qfwfq: That was what I was attempting to say too. Thanks for clarifying.