# Generic Sets and Category Theory

Hi,

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!

Gianluca

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Have you looked at chapter VI of Mac Lane & Moerdijk's Sheaves in Geometry and Logic? –  Finn Lawler Oct 8 '11 at 20:25
As a bridge, the topos-theoretic approach (as exposed in Mac Lane and Moerdijk) is spiritually closer to the Boolean-valued model approach of Scott and Solovay than it is to the original approach of Cohen, so a committed set theorist might prefer reviewing that first. –  Todd Trimble Oct 8 '11 at 20:45
The category of sets with arrows corresponding to subsets is a very large poset with far less structure than it should. With a little additional structure you get a ZF algebra, which is much more useful. (See Algebraic Set Theory by Joyal and Moerdijk.) –  François G. Dorais Oct 8 '11 at 22:20
(retracted previous comment) Reviewing my copy I noticed that the forcing technology setup in "Sheaves in Geometry and Logic" is more in line with the purely syntactic interpretation of forcing. Making it not very useful with regards to answering your question. –  Michael Blackmon Oct 9 '11 at 18:15