Obtaining a subcanonical topology

Question

Given a small site $(\mathbb{C}, \mathcal{J})$, it is known that one can devise a subcanonical site $(\mathbb{D}, \mathcal{K})$ so that $\mathsf{Sh}(\mathbb{C}, \mathcal{J}) \simeq \mathsf{Sh}(\mathbb{D}, \mathcal{K})$. Is it possible in general to choose $\mathbb{D} := \mathbb{C}$, i.e. is it possible to keep the same base category and just modify the topology? If not, then is there a general construction of $\mathbb{D}$ from $(\mathbb{C}, \mathcal{J})$ that is "close" to the category $\mathbb{C}$ (which is an admittedly vague question)?

Answer 1

No, it is not always possible to choose $\mathcal{D} = \mathcal{C}$. For example, suppose $J$ is the topology where every sieve – even the empty sieve – covers; then the topos is trivial, so any subcanonical site for it must have the property that there is exactly one morphism between any two objects. But $\mathcal{C}$ could be an arbitrary category.

In general, the way to obtain a subcanonical site from an arbitrary site is to take the full subcategory $\mathcal{D}$ of the topos spanned by the sheafification of the representable presheaves. The topology $K$ has as its covering sieves the sieves in $\mathcal{D}$ that are jointly epimorphic in the topos. By construction there is a bijective on objects functor $\mathcal{C} \to \mathcal{D}$ that sends $J$-covering sieves to $K$-covering sieves.

In fact, $J$ is subcanonical if and only if this $\mathcal{C} \to \mathcal{D}$ is fully faithful; and if $J$ is subcanonical then $J = K$, modulo the identification of $\mathcal{C}$ and $\mathcal{D}$. So you could think of this construction as "the best" subcanonical site related to the site you start with.