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).

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    $\begingroup$ Regarding your last question, there are basically no limits in $\mathbf{qCat_2}$. Instead you have weak limits in the sense Riehl and Verity study, such as the cotensor by the arrow. $\endgroup$ – Kevin Carlson Jun 30 '16 at 7:10
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    $\begingroup$ The Joyal model structure is enriched over itself, so homotopy (co)limits can be computed as derived weighted (co)limits with respect to the constant weight. Joyal cofibrations of simplicial sets coincide with monomorphisms, so a cofibrant replacement of the constant weight can be taken to be the same as for the Kan—Quillen model structure on simplicial sets. $\endgroup$ – Dmitri Pavlov Jun 30 '16 at 17:10
  • $\begingroup$ @DmitriPavlov, I have been a little confused by your comment. It is true that the Joyal model structure and the Quillen model structure have the same cofibrations (monomorphisms), but I don't see why this implies that we can use the same cofibrant replacement for the constant weight. After all, the two model structures certainly don't have the same acyclic cofibrations. That is why I suggested that we might have to take the free groupoid of $\mathcal{D}/-$ before taking the nerve (but I'm unsure if that's correct). $\endgroup$ – Kyle Ferendo Jul 7 '16 at 6:13
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    $\begingroup$ @KyleFerendo: I'm not sure where acyclic cofibrations are supposed to appear in your picture. A cofibrant replacement of some diagram X can be obtained by factoring ∅→X as a composition of a cofibration and an acyclic fibration. These two classes of maps coincide for Joyal and Kan—Quillen model structures. $\endgroup$ – Dmitri Pavlov Jul 7 '16 at 10:43
  • $\begingroup$ @DmitriPavlov Yes, thank you! Somehow I had tricked myself into thinking about fibrant replacement. You're completely correct. $\endgroup$ – Kyle Ferendo Jul 7 '16 at 16:18

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).

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    $\begingroup$ In case you are interested to the homotopy colimit of a diagram this is given by the fibrant replacement of the total category $\mathcal{C}^\natural$ in marked simplicial sets. Also, if it happens to be more convenient you can work with cartesian fibrations instead and the same theorems are true. $\endgroup$ – Denis Nardin Jun 30 '16 at 7:27
  • $\begingroup$ Thanks very much for the answer; this is the sort of thing I was looking for. I had been aware that Lurie used coCartesian fibrations to calculate limits within the quasicategory of quasicategories, but I did not think to apply the same approach to the actual category of quasicategories. Also, thanks for the clarifications about qCat2. I had thought there might be hope because results about adjunctions hold, but I'm going to give Riehl and Verity's joint work another read just so I know what's true and what's not about qCat2. $\endgroup$ – Kyle Ferendo Jun 30 '16 at 14:48

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