I am working on a project which requires that I calculate homotopy limits of homotopy theories (i.e. $(\infty,1)$-categories). It may be relevant that the homotopy limits which interest me are in the shape of towers; that is, the indexing category looks like $\cdots\rightarrow\cdot\rightarrow\cdot$. Because I am interested in some category-theoretic constructions (e.g. homotopy adjunctions between $(\infty,1)$-categories, Kan extensions, and homotopy (co)limits within $(\infty,1)$-categories), quasicategories seem like a natural choice for modeling $(\infty,1)$-categories, since they have the most well-developed category theory. However, I would be open to using a different model (e.g. Bergner's model structure on simplicial categories or Rezk's complete Segal spaces) if it proved more convenient.


Anyway, my question is simply how to calculate these homotopy limits. I am aware of Emily Riehl's proof that there is a model structure on the category of marked simplicial sets which is Quillen equivalent to the Joyal model structure on simplicial sets. This is nice, because the former model category is a simplicial model category, which the latter is not (although it is Cartesian closed). If it comes down to it, I should be able to do everything I want to do in this setting, bringing to bear the techniques developed in Riehl's book *Categorical Homotopy Theory* for calculating homotopy limits in simplicial model categories. But I'd like to know if there's a more straightforward approach which does not leave the Joyal model structure on simplicial sets.


Sticking with Riehl's approach to homotopy limits via weighted limits, is there a particular weight I should be using for calculating homotopy limits in $\mathbf{qCat\_\infty}$ as a simplicial category (but again, not a simplicial model category)? I don't know that $N(\mathcal{D}/-):\mathcal{D}\to\mathbf{sSet}\_\mathrm{Joyal}$ is cofibrant in $\left(\mathbf{sSet}\_\mathrm{Joyal}\right)\_\mathrm{proj}^\mathcal{D}$. Does anyone know if it is, or if not, what the cofibrant replacement looks like? Would we take the free groupoid of $\mathcal{D}/-$ before taking the nerve (just a guess)?


One last question: the theory developed by Riehl and Verity in their series of papers makes the 2-categorical approach to the study of $(\infty,1)$-categories appealing (i.e. working in the homotopy 2-category of the $(\infty,2)$-category of $(\infty,1)$-categories). Does anyone know if homotopy limits in $\mathbf{qCat\_\infty}$ agree with $\mathbf{Cat}$-enriched (conical) limits in $\mathbf{qCat\_2}$? That would be a useful result, but I don't think I've seen anything to that effect in Riehl's book.


Last note: one of the limits which interests me is of a tower of isofibrations, but I don't know that the morphisms involved in the second limit are inner fibrations (although the diagram is certainly pointwise fibrant).



Methods for defining/calculating homotopy limits of quasicategories
When working with quasi-categories, it is often more convenient (and more compatible with existing machinery) not to work with actual strict diagrams of quasi-categories but rather with **coCartesian fibrations**. In your case this would be a coCartesian fibration of the form $\pi:\mathcal{C} \to N(I)$ where $I$ is your indexing category (e.g., the tower category) and $N(I)$ is its nerve. The limit is then given explicitly as the quasi-category of coCartesian sections $N(I) \to \mathcal{C}$ of $\pi$ (that is, sections which send every edge of $N(I)$ to a $\pi$-coCartesian edge of $\mathcal{C}$). This is a sub-simplicial set of the mapping quasi-category cut out by straightforward degree-wise conditions, so it doesn't get more explicit than that. Of course, you need first to translate your diagram to a coCartesian fibration. In your case the indexing diagram is discrete so you can use the **relative nerve construction** featured in section 3.2.5 of higher topos theory (here you will again have an explicit formula for the simplices of $\mathcal{C}$, so all in all the end result will be explicit). It could also be that there is a better way to get a coCartesian fibration over $N(I)$ in your particular case. For example, it could be that you constructed your diagram from some initial data, but that you could in fact construct a coCartesian fibration from that initial data just as easily.


Concerning the possibility of working with the homotopy 2-category of $(\infty,1)$-categories, the latter most likely does not have (ordinary or $\mathrm{Cat}$-enriched) limits and colimits in general, no more than the the homotopy category of spaces has limits and colimits (though I admit I don't have a concrete counter-example in mind of a diagram in the homotopy 2-category which does not have a limit).


