## November 6, 2010

### Structures in a Cohesive ∞-Topos

#### Posted by Urs Schreiber

A cohesive $\infty$-topos is a big $\infty$-topos $\mathbf{H}$ 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 $(Disc \dashv \Gamma): \mathbf{H} \to \infty Grpd$ admits a further left adjoint $\Pi$ and a further right adjoint $CoDisc$:

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

with $Disc$ and $Codisc$ both full and faithful and such that $\Pi$ moreover preserves finite products.

The existence of such a quadruple of adjoint $\infty$-functors alone implies a rich internal higher geometry in $\mathbf{H}$ 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 $\mathbf{H}$, one would hope that there naturally is associated with it a geometry (for structured $\infty$-toposes) $\mathcal{G}$ such that for every concrete object $X \in \mathbf{H}$ the over-$\infty$-topos $\mathbf{H}/X$ is naturally a little $\mathcal{G}$-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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