Consider the nerve functor $N : \mathbf{Cat} \to \mathbf{sSet}$; it is fully faithful, preserves finite limits, products and exponentials, etc. I am wondering if it additionally preserves whatever dependent products exist in $\mathbf{Cat}$.

For instance, even though $\mathbf{Cat}$ is not locally Cartesian closed, we do have right adjoints to pullback along product projections. In this case, I'm curious whether the nerve functor takes these to dependent products in the category of simplicial sets.

If these dependent products are indeed preserved, I am interested in understanding the general conditions under which other "nerve-like" situations exhibit the same behavior, meaning functors of the form $X \mapsto \mathbb{D}[i -,X]$ for $i : \mathbb{C}\to \mathbb{D}$.

EDIT: I was hypothesizing that this property would follow from the following conditions:

1. the subcategory that generates the nerve is dense, and thence the nerve is fully faithful
2. the nerve functor has a left adjoint

Consider the nerve functor $N : \mathbf{Cat} \to \mathbf{sSet}$; it is fully faithful, preserves finite limits, products and exponentials, etc. I am wondering if it additionally preserves whatever dependent products exist in $\mathbf{Cat}$.

For instance, even though $\mathbf{Cat}$ is not locally Cartesian closed, we do have right adjoints to pullback along product projections. In this case, I'm curious whether the nerve functor takes these to dependent products in the category of simplicial sets.

If these dependent products are indeed preserved, I am interested in understanding the general conditions under which other "nerve-like" situations exhibit the same behavior, meaning functors of the form $X \mapsto \mathbb{D}[i -,X]$ for $i : \mathbb{C}\to \mathbb{D}$, primarily considering the case where $i$ is a subcategory inclusion.

EDIT: I was hypothesizing that this property would follow from the following conditions:

1. the subcategory that generates the nerve is dense, and thence the nerve is fully faithful
2. the nerve functor has a left adjoint

Consider the nerve functor $N : \mathbf{Cat} \to \mathbf{sSet}$; it is fully faithful, preserves finite limits, products and exponentials, etc. I am wondering if it additionally preserves whatever dependent products exist in $\mathbf{Cat}$.

For instance, even though $\mathbf{Cat}$ is not locally Cartesian closed, we do have right adjoints to pullback along product projections. In this case, I'm curious whether the nerve functor takes these to dependent products in the category of simplicial sets.

If these dependent products are indeed preserved, I am interested in understanding the general conditions under which other "nerve-like" situations exhibit the same behavior, meaning functors of the form $X \mapsto \mathbb{D}[i -,X]$ for $i : \mathbb{C}\to \mathbb{D}$.

Consider the nerve functor $N : \mathbf{Cat} \to \mathbf{sSet}$; it is fully faithful, preserves finite limits, products and exponentials, etc. I am wondering if it additionally preserves whatever dependent products exist in $\mathbf{Cat}$.

For instance, even though $\mathbf{Cat}$ is not locally Cartesian closed, we do have right adjoints to pullback along product projections. In this case, I'm curious whether the nerve functor takes these to dependent products in the category of simplicial sets.

If these dependent products are indeed preserved, I am interested in understanding the general conditions under which other "nerve-like" situations exhibit the same behavior, meaning functors of the form $X \mapsto \mathbb{D}[i -,X]$ for $i : \mathbb{C}\to \mathbb{D}$.

EDIT: I was hypothesizing that this property would follow from the following conditions:

1. the subcategory that generates the nerve is dense, and thence the nerve is fully faithful
2. the nerve functor has a left adjoint