I am interested in categorical versions of forcing techniques, and I was wondering if there is anyway that topological/set-theoretical concepts such as filters, density and generic sets can be defined in a categorical way. I am aware of the work that has been done by Tierney and Lawere, although I was looking for something somewhat more basic and, ultimately, more simple. In particular, if we consider the category of sets where the arrows are defined by the subset relations, can we define the concepts above in that category? And if so, what would they look like? I apologize in advance if the question is trivial or not very well-posed, but I am a categorical absolute beginner!