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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 10th 2014
• (edited Jun 10th 2014)

Larusson formulates the Oka principle homotopy-theoretically as: a complex manifold $X$ is Oka if for every Stein manifold $\Sigma$ the canonical map

$Maps_{hol}(\Sigma, X) \to Maps_{top}(\Sigma,X)$

between the mapping spaces is a weak homotopy equivalence (see here).

It is natural to wonder what this looks like in terms of the cohesion of the $\infty$-topos $\mathbb{C}Anlytic\infty Grpd$ over $CplxMfd$.

If we write $\Pi : \mathbb{C}Anlytic\infty Grpd \to \infty Grpd$, then up to possible technicalities to be checked, it should simply mean

$\Pi[\Sigma, X] \stackrel{\simeq}{\longrightarrow} [\Pi \Sigma,\; \Pi X]$

where $[-,-]$ is the internal hom.

(Something close to this (but not quite the same) is what Lawvere calls the “axiom of continuity” in a cohesive topos.)

If instead we work internally and let $\Pi : \mathbb{C}Anlytic\infty Grpd \to\mathbb{C}Anlytic\infty Grpd$ be the shape modality, then the above is equivalently

$\Pi[\Sigma, X] \stackrel{\simeq}{\longrightarrow} \flat [\Pi \Sigma,\; \Pi X] \,.$

In either case, it is a very natural condition to ask for in general cohesive $\infty$-toposes. Maybe one should call it the Oka-Larusson property or something…