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November 6, 2010

Structures in a Cohesive ∞-Topos

Posted by Urs Schreiber

A cohesive \infty-topos is a big \infty-topos H\mathbf{H} that provides a context of generalized spaces in which higher/derived geometry makes sense.

It is an \infty-topos whose global section \infty-geometric morphism (DiscΓ):HGrpd(Disc \dashv \Gamma): \mathbf{H} \to \infty Grpd admits a further left adjoint Π\Pi and a further right adjoint CoDiscCoDisc:

(ΠDiscΓCodisc):HGrpd (\Pi \dashv Disc \dashv \Gamma \dashv Codisc) : \mathbf{H} \to \infty Grpd

with DiscDisc and CodiscCodisc both full and faithful and such that Π\Pi moreover preserves finite products.

The existence of such a quadruple of adjoint \infty-functors alone implies a rich internal higher geometry in H\mathbf{H} that comes with its internal notion of Galois theory , Lie theory , differential cohomology and Chern-Weil theory .

In order of appearance:

So cohesive \infty-toposes should be a fairly good axiomatization of higher/derived geometry in big \infty-toposes. There is also an axiomatization of higher/derived geometry on little \infty-toposes, called structured \infty-toposes.

There ought to be a good way to connect the big and the little perspective on higher/derived geometry. For instance given a cohesive \infty-topos H\mathbf{H}, one would hope that there naturally is associated with it a geometry (for structured \infty-toposes) 𝒢\mathcal{G} such that for every concrete object XHX \in \mathbf{H} the over-\infty-topos H/X\mathbf{H}/X is naturally a little 𝒢\mathcal{G}-structured \infty-topos.

I have some ideas on this, but am not really sure yet.

Posted at November 6, 2010 2:04 PM UTC

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Tracked: November 7, 2010 7:25 PM

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