Vrftsjtryk

In the first paragraph of p.41 of Introduction to: classification theory for abstract elementary class, Shelah gives the following definition of (Galois) type.

For $Mpreceq N_ell$ and $a_ellin N_ell setminus M$, $mathbf {tp}(a_1,M,N_1)=mathbf{tp}(a_2,M,N_2)$ iff for some $preceq$-extension $N_3$ of $N_2$ there is a $preceq$-embedding of $N_1$ into $N_3$ over $M$ which maps $a_1$ to $a_2$, recalling that $mathfrak K_lambda$ has amalgamation.

I have omitted references in the notation to the AEC $mathfrak{K}_lambda$ in which this is taking place.

How do I see that this definition is equivalent to the following definition?

For $Mpreceq N_ell$ and $a_ellin N_ell setminus M$, $mathbf {tp}(a_1,M,N_1)=mathbf{tp}(a_2,M,N_2)$ iff there exist $preceq$-embeddings $h: N_1to N_3$ and $g:N_2to N_3$ such that the resulting square commutes and $h(a_1) = g(a_2)$.

Also:

1. Why does Shelah write "recalling that $mathfrak K_lambda$ has amalgamation" after his definition?




2. In the definition, why does Shelah write $mathbf{tp}_{mathfrak{K}_lambda}(a_1,M,N_1)=mathbf{tp}_{mathfrak{K}}(a_2,M,N_2)$? (Note: there is no $lambda$ in the r.h.s.,but there is one in the l.h.s.)




3. In the second definition, do we assume that $g$ and $h$ fix $M$?

These definitions are saying almost exactly the same thing. They really only differ in whether they require $N_2$ to be a substructure of $N_3$, or just embedded into it.

Suppose $mathbf{tp}(a_1,M,N_1) = mathbf{tp}(a_2,M,N_2)$ in the sense of the first definition. Then we have an extension $N_2preceq N_3$ and a $preceq$-embedding $hcolon N_1to N_3$ over $M$ such that $h(a_1) = a_2$. Taking $g$ to be the inclusion $N_2to N_3$, we find that the square commutes: for all $min M$, $h(m) = m = g(m)$. [This is what it means to say that $h$ is a map over $M$.] And $h(a_1) = a_2 = g(a_2)$. So $mathbf{tp}(a_1,M,N_1) = mathbf{tp}(a_2,M,N_2)$ in the sense of the second definition.

Conversely, suppose $mathbf{tp}(a_1,M,N_1) = mathbf{tp}(a_2,M,N_2)$ in the sense of the second definition. Then we have $preceq$-embeddings $hcolon N_1to N_3$ and $gcolon N_2to N_3$ such that the resulting square commutes and $h(a_1) = g(a_2)$. Now we may assume that $N_2preceq N_3$ and $g$ is the inclusion mapping, by replacing $N_3$ with an isomorphic copy. And then $h(a_1) = g(a_2) = a_2$, so $mathbf{tp}(a_1,M,N_1) = mathbf{tp}(a_2,M,N_2)$ in the sense of the first definition.

For your additional questions:

1. It's because a more complicated definition of Galois type turns out to be necessary in AECs without amalgamation. If the AEC lacks amalgamation, then the relation $text{tp}(a_1,M,N_1) = text{tp}(a_2,M,N_2)$ defined in the question (in either of the equivalent ways) might not be transitive, and hence not an equivalence relation on triples $(a,M,N)$. I believe the usual fix is just to take the transitive closure of this relation.


2. My best guess is that it's a typo.


3. I don't know what this means. We have $Msubseteq N_1$ and $hcolon N_1to N_3$. Since $M$ is not assumed to be a subset of $N_3$, $h$ can't fix $M$. Certainly definition 2 assumes that for all $min M$, $h(m) = g(m)$, which is exactly what it means to say the square commutes.