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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 $X$ 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…