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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 10th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexHow should one complete the analogy >hyperdoctrine : topos :: *** : $(\infty, 1)$-topos ? From [[hyperdoctrine]] >The notion of a hyperdoctrine is essentially an axiomatization of the collection of slices of a locally cartesian closed category (or something similar) So what is >essentially an axiomatization of the collection of slices of a locally cartesian closed $(\infty, 1)$-category (or something similar)? Presumably syntactically we need a base category of contexts, then fibred above the possible expressions in the variables of the context types.

   How should one complete the analogy

   hyperdoctrine : topos :: *** : $(\infty, 1)$-topos ?

   From hyperdoctrine

   The notion of a hyperdoctrine is essentially an axiomatization of the collection of slices of a locally cartesian closed category (or something similar)

   So what is

   essentially an axiomatization of the collection of slices of a locally cartesian closed $(\infty, 1)$-category (or something similar)?

   Presumably syntactically we need a base category of contexts, then fibred above the possible expressions in the variables of the context types.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 10th 2013
   • (edited Jul 10th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThere are clearly more canonically expert people here than me, but let me say something anyway: I'd think this is pretty obvious: a hyperdoctrine -- contrary to what the name might suggest -- is a rather simplistic concept: a functor to a 2-category such that each morphism in the image has left and right adjoint. Add extra conditions as need be, but this is the basic idea. And this has the evident homotopy-theoretic refinement: an $(\infty,2)$-functor to an $(\infty,2)$-category landing in morphisms that have left and right $\infty$-adjoint. And as in the 1-categorical case, the self-indexing of any locally cartesian closed $(\infty,1)$-category is an example.

   There are clearly more canonically expert people here than me, but let me say something anyway:

   I’d think this is pretty obvious: a hyperdoctrine – contrary to what the name might suggest – is a rather simplistic concept: a functor to a 2-category such that each morphism in the image has left and right adjoint. Add extra conditions as need be, but this is the basic idea.

   And this has the evident homotopy-theoretic refinement: an $(\infty,2)$-functor to an $(\infty,2)$-category landing in morphisms that have left and right $\infty$-adjoint.

   And as in the 1-categorical case, the self-indexing of any locally cartesian closed $(\infty,1)$-category is an example.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJul 10th 2013
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexWhat Urs said.

   What Urs said.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 10th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThanks. Great. I was wondering about this in a talk by [Yoshihiro Maruyama](http://researchmap.jp/ymaruyama/?lang=english) on [Monad-Relativized Hyperdoctrines](http://researchmap.jp/mug46aqn5-28320/). So maybe there could be a parallel route to the one to [[modal hyperdoctrines]] to one to modal homotopy type theory. But I must dash.

   Thanks. Great. I was wondering about this in a talk by Yoshihiro Maruyama on Monad-Relativized Hyperdoctrines. So maybe there could be a parallel route to the one to modal hyperdoctrines to one to modal homotopy type theory.

   But I must dash.