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1. • CommentRowNumber1.
   • CommentAuthorspitters
   • CommentTimeFeb 1st 2014
   • (edited Feb 1st 2014)
   • PermaLink
   Author: spitters
   Format: MarkdownItexI expect the usual classes of [geometric morphisms](http://ncatlab.org/nlab/show/%28infinity%2C1%29-geometric+morphism) (inclusion, surjection, open, ...) to have higher analogues. However, this does not seem to be in the nlab, or in Lurie. Where should I be looking?

   I expect the usual classes of geometric morphisms (inclusion, surjection, open, …) to have higher analogues. However, this does not seem to be in the nlab, or in Lurie. Where should I be looking?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 1st 2014
   • (edited Feb 1st 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSince typically the characterization of geometric morphisms makes sense verbatim also in $\infty$-topos theory, most entries don't distinguish between the two cases. For instance * [[base change geometric morphism]] discusses both cases in parallel. Also typically every kind of geometric morphism is the relative version of a corresponding type of topos, and currently the nLab entries are often named after the topos. For instance we have * [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]] * [[connected topos]] / [[∞-connected (∞,1)-topos]] * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]] * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]] * [[local topos]] / [[local (∞,1)-topos]] and each of these entries you may essentially also read as being about the corresponding geometric morphisms. (I think most of them discuss this.)

   Since typically the characterization of geometric morphisms makes sense verbatim also in $\infty$-topos theory, most entries don’t distinguish between the two cases.

   For instance

   • base change geometric morphism

   discusses both cases in parallel.

   Also typically every kind of geometric morphism is the relative version of a corresponding type of topos, and currently the nLab entries are often named after the topos.

   For instance we have

   • locally connected topos / locally ∞-connected (∞,1)-topos

     • connected topos / ∞-connected (∞,1)-topos

     • strongly connected topos / strongly ∞-connected (∞,1)-topos

     • totally connected topos / totally ∞-connected (∞,1)-topos

   • local topos / local (∞,1)-topos

   and each of these entries you may essentially also read as being about the corresponding geometric morphisms. (I think most of them discuss this.)