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I was reading a bit in Jech's 3rd ed. "Set Theory" and found the following description of the behaviour of limit stages in iterated forcing [Chapter 16, just after definition 16.29].

Let $\langle \Bbb P_\alpha,\dot{\Bbb Q}_\alpha\mid \alpha<\delta\rangle$ be an iteration of length $\delta$ where $\delta$ is a limit ordinal. Then $\Bbb P_\delta$ is called a direct limit when a $\delta$-sequence $p$ is a condition of $\Bbb P_\delta$ iff there is $\alpha<\delta$ such that $p\restriction \alpha\in \Bbb P_\alpha$ and $\Vdash_{\Bbb P_\xi}\text{"}p(\xi)=1_{\dot{\Bbb Q}_\xi}\text{"}$ for $\alpha\leq\xi<\delta$. On the other hand, $\Bbb P_\delta$ is called an inverse limit when a $\delta$-sequence $p$ is a condition of $\Bbb P_\delta$ iff $p\restriction \alpha\in \Bbb P_\alpha$ for all $\alpha<\delta$.

I understand the concepts mentioned, but I don't understand the use of this terminology. Could someone explain in what way these $\Bbb P_\delta$ are direct / inverse limits, preferably while avoiding category theory as much as possible? I have to admit that I have very little intuition of what an inverse limit is.

Vsotvep
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    Bounded support and full support would be (sort of) the corresponding definitions. Note that by Fodor's lemma, if your forcing is bounded on every limit stage, it is a finite support iteration. – Asaf Karagila Jun 17 '21 at 08:30
  • (I'll add that I recently told someone how much I hated that terminology when I was a student, and that it wasn't very clear to me what is meant by it.) – Asaf Karagila Jun 17 '21 at 08:53
  • @AsafKaragila what a coincidence! I did find a not-so-recent question of yours whose answers were helpful to me in understanding what an inverse limit is (more so than the Wikipedia page turned out to be): https://math.stackexchange.com/questions/38517/ – Vsotvep Jun 17 '21 at 12:43
  • Yes, I still don't fully understand that... Category theory is weird. :-P – Asaf Karagila Jun 17 '21 at 13:02

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In general terms, the direct limit should contain only as much information as is contained in the previous iterations. So in this sense, if we have something in the direct limit, it should really just be something in one of the previous iterations but translated to the full iteration just by adding a bunch of trivial entries at the end.

The inverse limit, on the other hand, is the largest thing that still counts as an iteration. It should have "projections" down to the previous iterations in the sense that things in the inverse limit should still be constructed from previous iterations, but there's basically no limit to what these constructions could be: as long as the initial segments are all conditions in previous iterations, then the full length sequence works. In a more practical sense, the "projections" used are just the restriction maps: $p\mapsto p\upharpoonright \alpha$ for each $\alpha$. In this way, the inverse limit is the largest thing that still counts as an iteration, i.e. we can restrict conditions down. In a similar phrasing, the direct limit is the smallest thing that still counts as an iteration.

In another way, if we think of conditions built up through the iterations as a kind of tree ($p\upharpoonright\alpha\lhd p\upharpoonright\beta$ for $\alpha<\beta$) then the inverse limit corresponds to taking all the cofinal branches of this tree whereas the direct limit corresponds to taking all the bounded branches of this tree (or rather, those with bounded support).

It's much easier to understand these conditions in terms of support however, and so Jech's definition, while not particularly motivated, is probably the most practical understanding to have. The mantra really is just that direct limits have bounded support whereas inverse limits use full support in the sense that a $\kappa$-length iteration with support in an ideal $I\subseteq\mathcal{P}(\kappa)$ is the inverse limit of previous iterations iff the support obeys $$\{x\in I:\sup(x)=\kappa\}=\{x\subseteq\kappa:\sup(x)=\kappa\wedge\forall\beta<\alpha\ (x\cap\beta\in I)\}\text{.}$$ And similarly, such an iteration is the direct limit of previous iterations iff the support obeys $$ I\subseteq\{x\subseteq\kappa:\sup(x)<\kappa\}\text{,}$$ equivalently, $I=\bigcup_{\beta<\alpha}I\cap \mathcal{P}(\beta)$.

In practice, the two are often very useful because it's relatively easy to confirm that a particular construction is indeed a condition in the iteration. With inverse limits, there's really no question: if it's a limit of conditions in previous iterations, it's a condition in the limit iteration regardless of what the support ends up looking like. With direct limits, you just need to show that the support is bounded, which is often pretty simple. So you often will see results in the literature related to iterations that use either direct or inverse limits at every limit stage, like Easton iterations.

JunderscoreH
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  • So, if I understand correctly, let $1=\langle 1_{\dot{\Bbb Q}\alpha}\mid \alpha<\delta\rangle$ be the trivial condition in $\Bbb P\delta$. If $\alpha\leq\beta$ we take a map $\mu_{\alpha\beta}:\Bbb P_\alpha\to \Bbb P_\beta$ sending $p_\alpha$ to $p_\alpha^{\frown}(1\restriction [\alpha,\beta))$. Under these maps we can reperesent the equivalence class of $p_\alpha\in \Bbb P_\alpha$ with $p=p_\alpha^{\frown}(1\restriction [\alpha,\delta))\in \Bbb P_\delta$, which makes $\mathrm{spt}(p)$ bounded. Then $\Bbb P_\delta$ a direct limit in the sense that I understand direct limits. – Vsotvep Jun 17 '21 at 12:34
  • On the other hand, for $\alpha\leq\beta$ we can take inverse (or projective) maps $\pi_{\beta\alpha}:\Bbb P_\beta\to\Bbb P_\alpha$ sending $p_\beta\mapsto p_\beta\restriction\alpha$. Then an element of the inverse limit is a sequence $\langle p_\xi\mid \xi<\delta\rangle$ such that for $\alpha\leq \beta$ we have $p_\alpha=\pi_{\beta\alpha}(p_\beta)$. We can represent these sequences as $\bigcup_{\xi\in{\delta}} p_\xi\in \Bbb P_\delta$, and these can have unbounded support. – Vsotvep Jun 17 '21 at 12:35
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    @Vsotvep Yes, so long as $\mathbb{P}\delta$ is the image of those $\mu{\alpha\delta}$s, it's the direct limit. I'll note that we always get these $\mu_{\alpha\beta}$s as embeddings from previous iterations into later ones, regardless of direct limit considerations. The fact that $\mathbb{P}_\delta$ is minimal with that property is what makes it the direct limit.

    And yes to the inverse limit characterization. Basically, it's the largest model that still works with previous iterations.

     – JunderscoreH Jun 17 '21 at 14:55