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).

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).

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).

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).