We all know that forcing can be seen (if you like things that way) as a category of sheaves over the poset of forcing conditions equipped with the double negation Grothendieck topology. As such it is a Grothendieck topos. We have several 'ready-made' notions of maps between such: logical morphisms and geometric morphisms. Now given a base category $Set$ of sets, it can be seen as sheaves over the point. And given some forcing poset $P$, we have a map of posets $P\to \ast$, hence morphisms between the base category of sets and the forced category of sets. This is all very standard, but only relates a forced category of sets to the base category.

How do we relate two forced categories of sets? This is one facet of David Corfield's recent question about the set-theoretic multiverse (itself inspired by JDH's paper on the same). One obvious way to do it is to come up with a category of forcing posets. Namely, if we are able to say what morphisms between posets (i.e. functors between the sites they define) should be allowed as 'forcing maps', then we automatically get the two notions of maps between the respective Grothendieck toposes aka forced categories of sets.

Can we get morphisms between forcing extensions this way?

complete embeddings. These are pretty well understood. Joel and others have worked out the modal theory of this. I remember working out the category-theoretic structure around 8-10 years ago... Since my organizational skills were lacking back then, I would probably need an archeological team to dig this up... (continued) $\endgroup$ – François G. Dorais♦ Sep 28 '11 at 2:37