What's new in the world of derived analytic geometry?? Let me mention some things I am aware of. I would be glad to hear from you also, in the comments below, maybe I missed something you have done or you have some other comment or question. First of all, some new preprints are out on the subject of dagger (or overconvergent) analytic geometry: http://arxiv.org/abs/1410.7971 by Frédéric Paugam. This paper gives a whirlwind tour of overconvergent function theory, and how it defines an "overconvergent stable homotopy theory of global analytic spaces" along with many other topics. Second of all, myself and Federico Bambozzi also have a preprint on the subject: http://arxiv.org/abs/1502.01401. The point here is that just like in ordinary algebraic geometry, in relative analytic geometry there is both a derived (or homotopy) and underived "ordinary" version. Most rings or algebras of functions that are familiar to algebraic geometers "live in degree zero" and one uses just standard algebra to manipulate them. The problem comes in the tensor product of two algebras over a third. This is not an exact functor and so we need to derive it. One can usually get away with just using the derived tensor product of modules. However, the result one gets is not an ordinary algebra. Therefore, in the derived world, there are new objects to be added. In analytic geometry, the algebras of functions are not finitely generated, they are usually tamed by adding some topology such as a Frechet structure. However, Frechet spaces do not form a closed symmetric monoidal category. Its better to use (complete, convex) bornological spaces. Back to the point I was trying to make, if you formulate analytic geometry as relative geometry, you have already done most of the work towards getting a good theory of derived analytic geometry. The hard part is showing that your theory agrees with standard notions in the literature. If you work with bornological algebras, you get a Grothendieck topology on this category which comes from the derived world. Now take any problem in standard analytic geometry where you want to construct some object satisfying a universal property. Its not clear what to do. However, since the category of bornological spaces has all limits and colimits, you can do what you wanted to in algebraic geometry relative to this category. This idea is due to Kobi Kremnizer. Read more about it in this article: http://arxiv.org/abs/1312.0338.

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