Methods for defining/calculating homotopy limits of quasicategories

Question

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).

Answer 1

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).