Re: "To get back to the original question, my understanding was that 
Voevodsky had done a bunch of scheme theory and it had got him a Fields 
medal and it was this mathematics which he was interested in at the time. 
He wanted to formalise his big theorem, just like Hales did."

I think he was more interested in formalizing things like his early work 
with Kapranov on higher categories, which turned out to have a mistake in 
it.  He once told me that he wasn't interested in formalizing his proof of 
Bloch-Kato, because he was sure it was right.  (I should have asked him at 
the time how he could be so sure!)

Re: "The clearest evidence, in my mind, is that there is no definition of a 
scheme in UniMath."

That's sort of accidental.   In early 2014, expecting to speak at an 
algebraic geometry in the summer, he mentioned that one idea he had for his 
talk would be to formalize the definition of scheme in UniMath and speak 
about it.  I think he was distracted from that by thinking about 
C-systems.  The UniMath project aims at formalizing all of mathematics, so 
the definition of scheme is one of the next things that (still) needs to be 
done.

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