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?

  • 4
    $\begingroup$ (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
  • 2
    $\begingroup$ 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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.