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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 24th 2017
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   Author: David_Corfield
   Format: MarkdownItexSince [evidently](https://nforum.ncatlab.org/discussion/7301/parametrized-higher-category-theory-and-higher-algebra/?Focus=65109#Comment_65109) an $(\infty, 1)$-version of profunctor is needed, I started [[(∞,1)-profunctor]] and [[(∞,1)Prof]]. Not sure if I 'homotopified' everything properly. Perhaps we can spark off a $(\infty, 1)$-[[equipment]]. It's in the [air](https://arxiv.org/abs/1607.04343) >What we would like to say now is that the ∞-category of inner fibrations $X \to S$ is equivalent to the ∞-category of lax functors from $S^{op}$ to a suitable “double ∞-category” of ∞-categories and profunctors. We do not know, however, how to make such an assertion precise.

   Since evidently an $(\infty, 1)$-version of profunctor is needed, I started (∞,1)-profunctor and (∞,1)Prof. Not sure if I ’homotopified’ everything properly.

   Perhaps we can spark off a $(\infty, 1)$-equipment. It’s in the air

   What we would like to say now is that the ∞-category of inner fibrations $X \to S$ is equivalent to the ∞-category of lax functors from $S^{op}$ to a suitable “double ∞-category” of ∞-categories and profunctors. We do not know, however, how to make such an assertion precise.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeAug 24th 2017
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   Author: Mike Shulman
   Format: MarkdownItexThanks! Did we really not have this before? I guess not.

   Thanks! Did we really not have this before? I guess not.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 24th 2017
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   Author: David_Corfield
   Format: MarkdownItexWe do have [[Pr(infinity,1)Cat]]. What exactly should be said about the relationship?

   We do have Pr(infinity,1)Cat. What exactly should be said about the relationship?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeAug 24th 2017
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   Author: Mike Shulman
   Format: MarkdownItex$(\infty,1)Prof$ should be equivalent to the full subcategory of $Pr(\infty,1)Cat$ whose objects are presheaf categories.

   $(\infty,1)Prof$ should be equivalent to the full subcategory of $Pr(\infty,1)Cat$ whose objects are presheaf categories.