The n-Category Café

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:

• $\infty$-Groups

• Cohomology

• Concordance

• Homotopy and Galois theory

• van Kampen theorem

• Paths

• Universal coverings and Whitehead towers

• Flat differential cohomology and local systems

• de Rham cohomology

• Concrete objects: Cohesive $\infty$-groupoids

• Infinitesimal objects: $\infty$-Lie algebroids

• Lie theory

• Maurer-Cartan forms and curvature characteristic forms

• Differential cohomology

• Chern-Weil theory

• Chern-Simons theory

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.