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1. • CommentRowNumber1.
   • CommentAuthornLab edit announcer
   • CommentTimeSep 2nd 2022
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexstarting discussion post with the advent of synthetic approaches to $(\infty,1)$-category theory like $\infty$-cosmos theory and simplicial type theory, there should be a definition of essentially surjective $(\infty,1)$-functor which does not make explicit reference to set-theoretic models. Anonymous <a href="https://ncatlab.org/nlab/revision/diff/essentially+surjective+%28infinity%2C1%29-functor/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/essentially+surjective+%28infinity%2C1%29-functor/6">v6</a>, <a href="https://ncatlab.org/nlab/show/essentially+surjective+%28infinity%2C1%29-functor">current</a>

   starting discussion post

   with the advent of synthetic approaches to $(\infty,1)$-category theory like $\infty$-cosmos theory and simplicial type theory, there should be a definition of essentially surjective $(\infty,1)$-functor which does not make explicit reference to set-theoretic models.

   Anonymous

   diff, v6, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 2nd 2022
   • (edited Sep 2nd 2022)
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   Author: Urs
   Format: MarkdownItexFor morphisms between $\infty$-groupoids, essential surjectivity is equivalent to (-1)-connectivity. So one needs the $(\infty,2)$-topos-version of $(-1)$-connectiveness, I'd think.

   For morphisms between $\infty$-groupoids, essential surjectivity is equivalent to (-1)-connectivity. So one needs the $(\infty,2)$-topos-version of $(-1)$-connectiveness, I’d think.

3. • CommentRowNumber3.
   • CommentAuthorHurkyl
   • CommentTimeSep 2nd 2022
   • (edited Sep 2nd 2022)
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexA functor between $\infty$-categories is essentially surjective iff the induced functor between their cores is an effective epimorphism. Isn't the reference to simplicially enriched categories is superfluous here? Since the point is to invoke the homotopy category construction $h : (\infty,1)Cat \to (1,1)Cat$ to reduce the question from $(\infty,1)$-categories to that of $(1,1)$-categories. I guess that passes the buck to whether you're happy with the definitions of $h$ and defining "essentially surjective" for (1,1)-categories

   A functor between $\infty$-categories is essentially surjective iff the induced functor between their cores is an effective epimorphism.

   Isn’t the reference to simplicially enriched categories is superfluous here? Since the point is to invoke the homotopy category construction $h : (\infty,1)Cat \to (1,1)Cat$ to reduce the question from $(\infty,1)$-categories to that of $(1,1)$-categories. I guess that passes the buck to whether you’re happy with the definitions of $h$ and defining “essentially surjective” for (1,1)-categories

4. • CommentRowNumber4.
   • CommentAuthorGuest
   • CommentTimeSep 2nd 2022
   • PermaLink
   Author: Guest
   Format: MarkdownItexSomebody like Mike Shulman or Emily Riehl might know more about the situation but I don't think anybody has formally defined the homotopy category construction in simplicial type theory yet. Don't know anything about infinity-cosmos theory so can't comment on that.

   Somebody like Mike Shulman or Emily Riehl might know more about the situation but I don’t think anybody has formally defined the homotopy category construction in simplicial type theory yet. Don’t know anything about infinity-cosmos theory so can’t comment on that.

5. • CommentRowNumber5.
   • CommentAuthornLab edit announcer
   • CommentTimeSep 2nd 2022
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexmentioning simplicial enriched categories is not needed as the construction should work for any model of $\infty$-category also added alternate definition mentioned in the discussion page using the core rather than homotopy categories. Anonymous <a href="https://ncatlab.org/nlab/revision/diff/essentially+surjective+%28infinity%2C1%29-functor/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/essentially+surjective+%28infinity%2C1%29-functor/7">v7</a>, <a href="https://ncatlab.org/nlab/show/essentially+surjective+%28infinity%2C1%29-functor">current</a>

   mentioning simplicial enriched categories is not needed as the construction should work for any model of $\infty$-category

   also added alternate definition mentioned in the discussion page using the core rather than homotopy categories.

   Anonymous

   diff, v7, current