It is well-known that the category of discrete fibrations over a category $\mathbb{C}$ is equivalent to the category of presheaves on $\mathbb{C}$.
More generally I think it is true, and probably well-known, though I can't find a reference, that the following two categories are equivalent:
- the category with objects: discrete fibrations $\mathbb{E} \to \mathbb{B}$ and morphisms: pairs of functors ($\mathbb{E}\to \mathbb{E}'$, $\mathbb{B} \to \mathbb{B}'$) making the square commute
- the category with objects: pairs ($\mathbb{B}$, $P_\mathbb{B}: \mathbb{B}^\mathrm{op} \to \mathbf{Set}$) consisting of a category and a presheaf on it, and morphisms: pairs $(f : \mathbb{B} \to \mathbb{B}', \alpha : P_\mathbb{B} \Rightarrow P_{\mathbb{B}'} \circ f^\mathrm{op})$ consisting of a functor and a natural transformation.
Suppose $\mathbb{B}$ is a site (i.e. has a topology), then the full subcategory of separated presheaves on $\mathbb{B}$ is a reflective subcategory of the category of all presheaves.
My question is:
Consider the variation of (2) above where the objects are ($\mathbb{B}, P_\mathbb{B}$) where $\mathbb{B}$ is a site. Call this $\mathbf{C}$.
Is it true that the full subcategory of $\mathbf{C}$ whose objects are ($\mathbb{B}, P_\mathbb{B}$) with $P_\mathbb{B}$ separated is a reflective subcategory of $\mathbf{C}$?
