Given a topology on a set $X$, let $2^X$ be the poset of subsets of $X$ ordered by inclusion. Then the interior operator for the topology is a comonad on $2^X$. In fact the topologies on $X$ correspond precisely to the finite-limit-preserving comonads on $2^X$. The coalgebras of the comonad are precisely the open sets.
Given a topological space $X$, define a bundle on $X$ to be a topological space $Y$ and a continuous map $f:Y\to X$. The category of bundles is the overcategory $\mathbf{Top}/X$. Say that a bundle is étalé if the map $f$ is a local homeomorphism. Then the étalé bundles form a coreflective subcategory of $\mathbf{Top}/X$, meaning that there is an étalification comonad on $\mathbf{Top}/X$. The coalgebras of the comonad are precisely the étalé bundles, which correspond to sheaves on $X$.
The first example is a special case of the second, in the sense that if we view a subset of $X$ with its inclusion map as a bundle then its étalification is precisely its interior. Note also that in the first example the coalgebras form a locale, and in the second they form a topos.