Fixing and pointwise-fixing of a structure in a suqare
$begingroup$
Suppose we have this square in the context of Abstract Elementary Class ${frak K}=(K,leq_frak K)$:
$$
require{AMScd}
begin{CD}
N_1 @>f_1>> N_3 \
@A{preceq_frak K}AA @AA{f_2}A\
M_0 @>{preceq_frak K}>> N_2
end{CD}
$$
My question is, what is the relationship among these properties of that square:
(1) the square commutes,
(2) the square fixes $M_0$, and
(3) the square fixes $M_0$ pointwise.
Moreover, what needs to be assumed about $M_0,N_1,N_2,N_3,f_1$ and $f_2$ to make sense of this question in each item (1),(2) and (3) above, respectivelly,besides that $f_1$ and $f_2$ are $preceq_frak K$-embeddings in $frak K$?
I'm esp. interested in the difference between (2) and (3). I also think
that (1) is the same as (3),but (2) is different.In fact, I'm not sure what (2) means and how it is different from (3).
logic category-theory first-order-logic predicate-logic model-theory
asked Jan 15 at 16:42
user122424user122424
1,1162716
$endgroup$
add a comment |
$begingroup$
Suppose we have this square in the context of Abstract Elementary Class ${frak K}=(K,leq_frak K)$:
$$
require{AMScd}
begin{CD}
N_1 @>f_1>> N_3 \
@A{preceq_frak K}AA @AA{f_2}A\
M_0 @>{preceq_frak K}>> N_2
end{CD}
$$
My question is, what is the relationship among these properties of that square:
(1) the square commutes,
(2) the square fixes $M_0$, and
(3) the square fixes $M_0$ pointwise.
Moreover, what needs to be assumed about $M_0,N_1,N_2,N_3,f_1$ and $f_2$ to make sense of this question in each item (1),(2) and (3) above, respectivelly,besides that $f_1$ and $f_2$ are $preceq_frak K$-embeddings in $frak K$?
I'm esp. interested in the difference between (2) and (3). I also think
that (1) is the same as (3),but (2) is different.In fact, I'm not sure what (2) means and how it is different from (3).
logic category-theory first-order-logic predicate-logic model-theory
asked Jan 15 at 16:42
user122424user122424
1,1162716
$endgroup$
$begingroup$
Your commutative diagram isn't compiling.
$endgroup$
– Kevin Carlson
Jan 15 at 16:43
$begingroup$
@KevinCarlson I can see that it isn't but I do not know why. While writting the question it was OK, but not now.
$endgroup$
– user122424
Jan 15 at 16:52
$begingroup$
@KevinCarlson Can you see the right version now?
$endgroup$
– user122424
Jan 15 at 17:02
1
$begingroup$
To me, the phrase "The square fixes $M_0$" (pointwise or not) is meaningless. You can talk about a particular map of sets fixing a set (pointwise), but not a square.
$endgroup$
– Alex Kruckman
Jan 15 at 17:44
$begingroup$
OK. What is the difference then fixing $A$ $f:Mto N$ for $Asubseteq |M|$ pointwise from fixing $A$ not pointwise?
$endgroup$
– user122424
Jan 15 at 18:14
add a comment |
$begingroup$
Suppose we have this square in the context of Abstract Elementary Class ${frak K}=(K,leq_frak K)$:
$$
require{AMScd}
begin{CD}
N_1 @>f_1>> N_3 \
@A{preceq_frak K}AA @AA{f_2}A\
M_0 @>{preceq_frak K}>> N_2
end{CD}
$$
My question is, what is the relationship among these properties of that square:
(1) the square commutes,
(2) the square fixes $M_0$, and
(3) the square fixes $M_0$ pointwise.
Moreover, what needs to be assumed about $M_0,N_1,N_2,N_3,f_1$ and $f_2$ to make sense of this question in each item (1),(2) and (3) above, respectivelly,besides that $f_1$ and $f_2$ are $preceq_frak K$-embeddings in $frak K$?
I'm esp. interested in the difference between (2) and (3). I also think
that (1) is the same as (3),but (2) is different.In fact, I'm not sure what (2) means and how it is different from (3).
logic category-theory first-order-logic predicate-logic model-theory
asked Jan 15 at 16:42
user122424user122424
1,1162716
$endgroup$
Suppose we have this square in the context of Abstract Elementary Class ${frak K}=(K,leq_frak K)$:
$$
require{AMScd}
begin{CD}
N_1 @>f_1>> N_3 \
@A{preceq_frak K}AA @AA{f_2}A\
M_0 @>{preceq_frak K}>> N_2
end{CD}
$$
My question is, what is the relationship among these properties of that square:
(1) the square commutes,
(2) the square fixes $M_0$, and
(3) the square fixes $M_0$ pointwise.
Moreover, what needs to be assumed about $M_0,N_1,N_2,N_3,f_1$ and $f_2$ to make sense of this question in each item (1),(2) and (3) above, respectivelly,besides that $f_1$ and $f_2$ are $preceq_frak K$-embeddings in $frak K$?
I'm esp. interested in the difference between (2) and (3). I also think
that (1) is the same as (3),but (2) is different.In fact, I'm not sure what (2) means and how it is different from (3).
logic category-theory first-order-logic predicate-logic model-theory
logic category-theory first-order-logic predicate-logic model-theory
asked Jan 15 at 16:42
user122424user122424
1,1162716
asked Jan 15 at 16:42
user122424user122424
1,1162716
edited Jan 15 at 17:01
user122424
asked Jan 15 at 16:42
user122424user122424
1,1162716
asked Jan 15 at 16:42
user122424user122424
1,1162716
asked Jan 15 at 16:42
user122424user122424
1,1162716
1,1162716
$begingroup$
Your commutative diagram isn't compiling.
$endgroup$
– Kevin Carlson
Jan 15 at 16:43
$begingroup$
@KevinCarlson I can see that it isn't but I do not know why. While writting the question it was OK, but not now.
$endgroup$
– user122424
Jan 15 at 16:52
$begingroup$
@KevinCarlson Can you see the right version now?
$endgroup$
– user122424
Jan 15 at 17:02
1
$begingroup$
To me, the phrase "The square fixes $M_0$" (pointwise or not) is meaningless. You can talk about a particular map of sets fixing a set (pointwise), but not a square.
$endgroup$
– Alex Kruckman
Jan 15 at 17:44
$begingroup$
OK. What is the difference then fixing $A$ $f:Mto N$ for $Asubseteq |M|$ pointwise from fixing $A$ not pointwise?
$endgroup$
– user122424
Jan 15 at 18:14
add a comment |
$begingroup$
Your commutative diagram isn't compiling.
$endgroup$
– Kevin Carlson
Jan 15 at 16:43
$begingroup$
@KevinCarlson I can see that it isn't but I do not know why. While writting the question it was OK, but not now.
$endgroup$
– user122424
Jan 15 at 16:52
$begingroup$
@KevinCarlson Can you see the right version now?
$endgroup$
– user122424
Jan 15 at 17:02
1
$begingroup$
To me, the phrase "The square fixes $M_0$" (pointwise or not) is meaningless. You can talk about a particular map of sets fixing a set (pointwise), but not a square.
$endgroup$
– Alex Kruckman
Jan 15 at 17:44
$begingroup$
OK. What is the difference then fixing $A$ $f:Mto N$ for $Asubseteq |M|$ pointwise from fixing $A$ not pointwise?
$endgroup$
– user122424
Jan 15 at 18:14
$begingroup$
Your commutative diagram isn't compiling.
$endgroup$
– Kevin Carlson
Jan 15 at 16:43
$begingroup$
Your commutative diagram isn't compiling.
$endgroup$
– Kevin Carlson
Jan 15 at 16:43
$begingroup$
@KevinCarlson I can see that it isn't but I do not know why. While writting the question it was OK, but not now.
$endgroup$
– user122424
Jan 15 at 16:52
$begingroup$
@KevinCarlson I can see that it isn't but I do not know why. While writting the question it was OK, but not now.
$endgroup$
– user122424
Jan 15 at 16:52
$begingroup$
@KevinCarlson Can you see the right version now?
$endgroup$
– user122424
Jan 15 at 17:02
$begingroup$
@KevinCarlson Can you see the right version now?
$endgroup$
– user122424
Jan 15 at 17:02
1
1
$begingroup$
To me, the phrase "The square fixes $M_0$" (pointwise or not) is meaningless. You can talk about a particular map of sets fixing a set (pointwise), but not a square.
$endgroup$
– Alex Kruckman
Jan 15 at 17:44
$begingroup$
To me, the phrase "The square fixes $M_0$" (pointwise or not) is meaningless. You can talk about a particular map of sets fixing a set (pointwise), but not a square.
$endgroup$
– Alex Kruckman
Jan 15 at 17:44
$begingroup$
OK. What is the difference then fixing $A$ $f:Mto N$ for $Asubseteq |M|$ pointwise from fixing $A$ not pointwise?
$endgroup$
– user122424
Jan 15 at 18:14
$begingroup$
OK. What is the difference then fixing $A$ $f:Mto N$ for $Asubseteq |M|$ pointwise from fixing $A$ not pointwise?
$endgroup$
– user122424
Jan 15 at 18:14
add a comment |
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$begingroup$
Your commutative diagram isn't compiling.
$endgroup$
– Kevin Carlson
Jan 15 at 16:43
$begingroup$
@KevinCarlson I can see that it isn't but I do not know why. While writting the question it was OK, but not now.
$endgroup$
– user122424
Jan 15 at 16:52
$begingroup$
@KevinCarlson Can you see the right version now?
$endgroup$
– user122424
Jan 15 at 17:02
1
$begingroup$
To me, the phrase "The square fixes $M_0$" (pointwise or not) is meaningless. You can talk about a particular map of sets fixing a set (pointwise), but not a square.
$endgroup$
– Alex Kruckman
Jan 15 at 17:44
$begingroup$
OK. What is the difference then fixing $A$ $f:Mto N$ for $Asubseteq |M|$ pointwise from fixing $A$ not pointwise?
$endgroup$
– user122424
Jan 15 at 18:14