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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 10th 2014
   • (edited Jun 10th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexLarusson 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](http://ncatlab.org/nlab/show/complex+analytic+∞-groupoid#OkaPrinciple)). It is natural to wonder what this looks like in terms of [the cohesion](http://ncatlab.org/nlab/show/complex+analytic+∞-groupoid#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...

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

   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 -topos over .

   If we write , then up to possible technicalities to be checked, it should simply mean

   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 be the shape modality, then the above is equivalently

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