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<p>Since Mike's thread <a href="http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=679&page=1">questions on structured (oo,1)-topos</a> got a bit highjacked by general oo-stack questions, I thought I'd start this new thread to announce attempted answers:</p>
<p>Mike had rightly complained in a query box that a "remark" of mine in which I had meant to indicate the intuitive meaning of the technical condition on an (oo,1)-structure sheaf had been "ridiculous".</p>
<p>But that condition is important, and important to understand. I have now removed the nonsensical paragraph and Mike's query box complaining about it, inserted a new query box saying "second attempt" and then spelled out two archetypical toy examples in detail, that illustrate what's going on.</p>
<p>The second of them can be found in StrucSp itself, as indicated. It serves mainly to show that an ordinary ringed space has a structure sheaf in the sense of structured oo-toposes precisely if it is a <a href="http://ncatlab.org/nlab/show/locally+ringed+space">locally ringed space</a>.</p>
<p>But to try to bring out the very simple geometry behind this even better, I preceded this example now by one where a structure sheaf just of continuous functions is considered.</p>
<p>Have a look.</p>
Okay, that makes more sense. It looks to me like a "geometry" is just another way of specifying the syntactic category of a geometric theory (suitably oo-ized). Thinking like that makes me feel like maybe I understand this business of "quantities" more. Maybe when I get a chance I'll try to write something connecting up this viewpoint.
It looks to me like a "geometry" is just another way of specifying the syntactic category of a geometric theory (suitably oo-ized).
I believe that's the kind of statement that I was thinking of when writing that remaining comment at classifying topos on the oo-version of "limit- and cover-preserving functors". I'd be very interested to see you pin this down more concretely. I feels like a very nice kind of thing, generally.
Is this choice of geometry anyhow related to the role of doctrines in algebraic geometry as analysed recently by Dolan (there are 6 videos online and John wrote osmething on his pages and there was a cafe discussion) ?
@Zoran, I would say that the "geometries" Lurie is using are all theories in the doctrine of left-exact idempotent-complete (oo,1)-sites. If you want to use a weaker doctrine, such as merely left-exact (oo,1)-categories, or even with merely finite products, you can "extend up" with the left adjoint, e.g. idempotent-completion, or the free left-exact category on a category with finite products. If you want to use a stronger doctrine, such as all small limits, then I think you're out of luck unless you generalize the definition.
By the way, when John started talking about this on the blog, I said "Did you compare this with what Lurie's written?" He never replied to that. I wonder if Jim Dolan is aware of it.
I added Kaledin's and Getzler-Kapranov refs to cyclic cohomology. GK paper is intriguing: there are two kinds of cyclic cohomology for an algebra over a cyclic operad HA called cyclic and HC which is a bit different. The Connes long exact sequence involves HA HH and HC, but HC is applied to an algebra over a Koszul dual operad, so if we have an associative algebra then the Koszul dual of the associative operad is again the associative operad, and HA and HC are basically undistinguishable, hence only HC. For general cyclic operad this is not so. I hope the operad enthusiasts from operadic kitchen Utrecht will soon teach us how to think of cyclic operads and HA vs HC. Sorry, I now see that I posted into wrong thread, but let it stay here then.
I hope the operad enthusiasts from operadic kitchen Utrecht will soon teach us how to think of cyclic operads and HA vs HC.
Okay, good hint. I'll see what the kitchen can do.
There is a rewarding goody at the next corner: so called Feynman transform for modular operads (which are a bit more than cyclic operads and of course have to do with CFT in special cases).
right, Simon Willerton was telling me about Feynman transforms the other day. I thought it was a pity that I didn't know about it.
but I need to organize this a bit, otherwise I'll get all the tasks that I need to.
Will the cyclic operads help me get a very general notion of Chern character? :-)
Probably not.
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