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The idea of a projection $\pi\colon\mathbb{Q}\to\mathbb{P}$ of forcing notions is something like a combinatorial stand-in for the fact that forcing with $\mathbb{Q}$ produces a generic for $\mathbb{P}$. One then studies the necessary quotient forcing to get your from $V^{\mathbb{P}}$ to $V^{\mathbb{Q}}$ and hopes it isn't too bad for whatever it is that needs to happen. This type of argument is typical when dealing with Mitchell- or Prikry-like forcings.

Unfortunately, as far as I'm able to tell, a lot of the facts about projections are relegated to folklore and rarely stated carefully. Even the definition of a projection seems to be a slippery thing! For example, here is the definition that most sources I've consulted agree on:

An order-preserving map $\pi\colon\mathbb{Q}\to\mathbb{P}$ is a projection if

  1. $\pi(1_{\mathbb{Q}})=1_{\mathbb{P}}$ and
  2. for any $q\in\mathbb{Q}$ and any $p\leq \pi(q)$ there is $\bar{q}\leq q$ such that $\pi(\bar{q})\leq p$.

This definition appears, for example, in Cummings' chapter in the Handbook of Set Theory. However, in a different chapter of the same Handbook, Abraham gives a definition which strengthens (2) above to $\pi(\bar{q})=p$. Abraham also mentions a third, even stronger version of the definition.

It also seems that most textbooks/references prefer to talk about complete suborders rather than projections (this is what Kunen does, as well as Shelah in Proper and Improper Forcing). Complete suborders are closely related to projections, but the two concepts are distinct and the exact relationship isn't totally clear to me.

Is there a canonical writeup of what a forcing projection is, along with a list (preferably with proofs) of their key properties? Ideally, this would include a spelled out connection between $\mathbb{Q}$ adding a generic for $\mathbb{P}$ and a projection of $\mathbb{Q}$ to something like the Boolean completion of $\mathbb{P}$, as well as a discussion of quotient forcing.

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  • $\begingroup$ Have you looked at the notes on my site? The part about projections is kinda early because a lot of students pestered me about it, but it's there. $\endgroup$
    – Asaf Karagila
    Commented Mar 12 at 8:08
  • $\begingroup$ @AsafKaragila Thanks for the pointer! I had a look, and you and Monroe do a good job presenting this stuff. I'm a bit reluctant to use it as a citation, but maybe I'll get over it. $\endgroup$ Commented Mar 15 at 16:03
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    $\begingroup$ If it's any consolation, the idea is to give another one or two of these courses and then compile it all into a book. $\endgroup$
    – Asaf Karagila
    Commented Mar 15 at 16:30

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I don’t know if there’s a “canonical” writeup, but I taught a master’s course a few years ago and wrote up many details of these things here. But maybe this isn’t useful if you’re looking for something to cite since it’s unpublished.

Regarding the Cummings/Abraham discrepancy, Cummings’ definition is better because it is more general. But situations satisfying Abraham’s definition are quite common, and some combinatorial arguments use that equality condition, so we shouldn’t forget it. In my experience, each of the various approaches to forcing and subforcing has some context where it looks like the most useful and elegant one.

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    $\begingroup$ Thanks for pointing me to your notes. It's helpful (especially the point that for Boolean algebras, the projection and regular embedding aspects match up!), although you're right that I'm reluctant to use it for a citation. Coming back to the difference between Cummings' and Abraham's definitions, I can cook up examples of Cummings projections that are not Abraham projections. Do you have an example of a Cummings projection $Q\to P$ where no Abraham projection exists? $\endgroup$ Commented Mar 15 at 16:01
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    $\begingroup$ @MihaHabič Yes, here’s one kind of squirrelly example. Abraham projections are surjective. Let P be some poset whose Boolean completion is strictly larger (like adding one Cohen real). Then the standard embedding of P into its completion is a Cummings projection, but there’s no surjection. $\endgroup$ Commented Mar 15 at 16:23
  • $\begingroup$ That makes sense. I had a follow-up question about the differences between the two projections and equivalent forcing notions, but I answered it for myself: if $\pi\colon Q\to P$ is a Cummings projection, then $\pi$ is an Abraham projection onto a dense subset of $P$. So the two notions of projection are equivalent if we are free to vary the codomain. $\endgroup$ Commented Mar 26 at 15:53
  • $\begingroup$ @MihaHabič Interesting. What is the dense subset? Is it just the image $\pi[Q]$? $\endgroup$ Commented Mar 26 at 20:24
  • $\begingroup$ Yes, (2) from the definition basically says that for any $q\in Q$ the image of the cone below $q$ is dense below $\pi(q)$. Then use $\pi(1)=1$. $\endgroup$ Commented Mar 26 at 22:38

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