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Actually, the closure operator of a topology is a finite colimit preserving monad on a powerset.

I'm not quite sure what you're asking, but under one way to interpret the question, an answer is that the theory of locales is a well-developed alternative to the classical theory of topological space …

Suppose $\mathscr{C}$ is the stackification of an internal category $C$ in $\mathbf{X}$. In this case, since $\mathrm{Sets}(\mathbf{X})$ is a stack, morphisms of stacks $\mathscr{C}^{\mathrm{op}} \to …

The other answers are all good, but I thought I would also point out that one doesn't have to require that sites be small, or have small dense sub-sites, or satisfy WISC. I think one does generally w …

First of all, Andreas' comment is right: a coverage gives no specified way to "pull back" a covering family of $U$ to a covering family of $V$. However, if you consider what Sketches of an Elephant c …

If by "topology" you mean the usual notion of "Grothendieck topology" (and not something weaker like "Grothendieck pretopology"), then the answer is yes. In fact, if $C$ is any site and $p\colon U\to …