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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?

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Having asked this question can I assume the content of Definition 7.12 (K. Kunen, "Set Theory: An Introduction to Independence Proofs", Chapter 7, Section 7, p. 222.) is not sufficient for your purposes? –  Michael Blackmon Sep 28 '11 at 2:21
    
IANAST (ST=Set Theorist). I've likewise seen lots of questions on MO about category theory for which the answer could be 'isn't this paper by Lawvere/Mac Lane/Benabou/Street/Baez/Lurie sufficient for you?' :) That doesn't mean they shouldn't have been asked. –  David Roberts Sep 28 '11 at 2:33
    
The natural notion of morphism between forcing posets in a given universe are known as 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) –  François G. Dorais Sep 28 '11 at 2:37
    
(continued) I do clearly remember the end of that story though. I thought that would make a good paper, but I was very young and I stumbled upon a FOM post of Bob Solovay that suggested that he had worked out all of that stuff and even more. I was very intimidated and I put all of that stuff in a folder, where it still lies assuming it hasn't been lost... As far as I can remember, nobody has published that kind of stuff anywhere, though it's kind of well-known to select people. Some papers by Blass and Scedrov do contain a lot of that stuff, but their goals were somewhat different. –  François G. Dorais Sep 28 '11 at 2:45
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François, would it be possible for you to post an answer explaining some of this in more detail? I agree with your remark that set theorists would understand the connection between forcing notions via complete embeddings, dense embeddings, projections, etc., but I would be interested to learn more about the category-theoretic and topological approaches you mention. –  Joel David Hamkins Oct 13 '11 at 13:57

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