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<p>I created <a href="https://ncatlab.org/nlab/show/synthetic+differential+geometry+applied+to+algebraic+geometry">synthetic differential geometry applied to algebraic geometry</a> which is supposed to host a question that I am going to post on <a href="http://go2.wordpress.com/?id=725X1342&site=sbseminar.wordpress.com&url=http%3A%2F%2Fmathoverflow.net%2F">math Overflow</a> following the discussion we have of that <a href="http://sbseminar.wordpress.com/2009/10/14/math-overflow/#comment-6875">here at SBS</a>.</p>
<p>In that context I also wrote a section at <a href="https://ncatlab.org/nlab/show/algebraic+geometry">algebraic geometry</a> intended to describe the general-nonsense perspective. But that didn't quite find the agreement with Zoran and while we are having some discussion about this in private, he has restructured that entry now.</p>
Added to synthetic differential geometry applied to algebraic geometry a short remark on functions $\mathbb{A}^1 \to \mathbb{A}^1$ internal to the big Zariski topos of a scheme: They are all polynomial functions. If people are interested, I can add the (routine) proof of this fact.
What exactly does that mean? I guess a second-order statement of the form $\forall f : \mathbb{A}^1 \to \mathbb{A}^1 . \exists n : \mathbb{N} . \exists a : \mathbb{N} \to \mathbb{A}^1 . \cdots$?
Yes, exactly. For me, $\forall f : \mathbb{A}^1 \to \mathbb{A}^1$ is, in the context of the internal language of a topos, syntactic sugar for $\forall f : [\mathbb{A}^1,\mathbb{A}^1]$, where $[\mathbb{A}^1,\mathbb{A}^1]$ is the internal Hom.
Note also, since the big Zariski topos is cocomplete, there is no difference in quantifying over the internal natural numbers ($\exists n : \mathbb{N}$) or taking a disjunction indexed by the external natural numbers ($\bigvee_{n \in \mathbb{N}}$).
I added this clarification to the article; thanks!
Thanks. I had completely forgotten about the existence of this entry. It’s been a long time. This ought to be entirely reworked. For the moment I have at least rearranged somewhat. Also cross-linked with the entry synthetic differential geometry itself.
Then I have copied the proposition over to affine line (in a new section Properties – Internal formulation), where it is more likely to be found by people who might care.
I think it would be good for these entries if the proof were indicated with the statement, yes.
Thanks Urs! I will add a proof today or tomorrow.
The axioms of synthetic differential geometry are intended to pin down the minimum general abstract axioms necessary for talking about the differential aspect of differential geometry using concrete objects that model infinitesimal spaces.
Shouldn’t this be modified now in view of the lesser structure presumed by differential cohesion?
That’s among the aspects that I was thinking of in #5. This entry was created over 5 years ago stating questions that I had back then; but meanwhile I had completely forgotten about the entry. Now I’d be inclined to rewrite it completely. But no time right now.
Added to affine line#InternalFormulation the proof promised in #6.
Thanks!
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