Under-site like subspace topology

Question

Let $X$ be a topological space and $Op(X)$ the category of its open sets. It is well known that $Op(X)$ has a canonical Grothendieck topology which makes of it a site. Let $U\in Op(X)$ be an object (an open set) of this category. It is naturally a topological space for the subspace topology. Hence there is a site $Op(U)$. I'd like to know if we can generalize this construction.

Let $(\mathcal{C},J)$ be a site and $c\in C$ an object. Can we talk of the "under-site" generated by $c$ ? It should be a good generalization of the subspace topology. What subcategory $\mathcal{C}_c$ of $\mathcal{C}$ should I consider ?

Answer 1

You want to take a slice category, instead of a subcategory. This is what you're actually doing in the classical example you give-it's just obscured by the fact that the site of a space is a poset. $\mathcal C/ x$, for any object $x$, has a natural site structure in which coverings are created by the forgetful functor to $\mathcal C$.