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I have two questions about images of morphisms of quasi-categories.

Suppose that $f\colon X \to Y$ is a morphism of quasi-categories.​

  1. ​Suppose that we calculate the image of $f$ in the category $\mathsf{QCat}$, using the universal property of image. Will this be a Joyal-fibrant replacement of the image as calculated in $\mathsf{SSet}$?
  2. If $X = N(C)$ and $Y = N(D)$ are (nerves of) categories, will the image of $f$ be the nerve of the essential image of the functor $C \to D$?

In case the answer to one of these questions is no, can the statements be adjusted to become true?

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    $\begingroup$ (I'm going to assume that with $\mathrm{QCat}$ you mean the 1-category of quasicategories) The answer to question 2 is no, for the same reason for which the image of a functor of categories is not the essential image: you are missing the objects that are equivalent to objects in the image but not in the image. I don't understand why you are taking the image in $\mathrm{QCat}$ though, it's not typically a useful thing to do. Maybe you can explain a bit your motivation? $\endgroup$ – Denis Nardin Nov 27 '18 at 10:36
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    $\begingroup$ Does a 1-categorical image in $QCat$ even exist? $\endgroup$ – Mike Shulman Nov 27 '18 at 18:32
  • $\begingroup$ I guess if you identify two $2$-simplices along their boundary (and fibrant replace), then there is no least sub-quasicategory containing the 1-skeleton, since the image of each $2$-simplex gives a sub-quasicategory. There is in fact no sub-quasicategory Joyal equivalent to the image, since every triangle commutes. $\endgroup$ – Kevin Carlson Nov 27 '18 at 19:54

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