What is the theorem being mentioned here?
$begingroup$
In this video, at just after the 5 minute mark, the speaker says:
"...the colimit of the diagram has the homotopy type of the homotopy colimit. Why? Because all of the maps included in the diagram are closed cofibrations, and there is a theorem that if you have a diagram where everything is a closed cofibration, the homotopy type of the colimit and the homotopy colimit are the same."
What is the theorem being referenced here, or alternatively how does this follow?
EDIT:
If I have the following diagram:
$$* rightrightarrows *$$
I think that this is a diagram of closed cofibrations for which the statement does not hold, for example.
I think the speaker's statement does hold in the particular type of cases that he is talking about, and that the maps in the diagrams he speaks of are closed cofibrations, but that there are extra conditions on the diagrams which he is not mentioning. I would like any further insight that anyone can offer (even in general, about diagrams whose colimits and hocolimits are homotopy equivalent).
algebraic-topology category-theory homotopy-theory polyhedral-product
asked Dec 6 '18 at 2:47
MattMatt
2,577719
$endgroup$
|
show 5 more comments
$begingroup$
In this video, at just after the 5 minute mark, the speaker says:
"...the colimit of the diagram has the homotopy type of the homotopy colimit. Why? Because all of the maps included in the diagram are closed cofibrations, and there is a theorem that if you have a diagram where everything is a closed cofibration, the homotopy type of the colimit and the homotopy colimit are the same."
What is the theorem being referenced here, or alternatively how does this follow?
EDIT:
If I have the following diagram:
$$* rightrightarrows *$$
I think that this is a diagram of closed cofibrations for which the statement does not hold, for example.
I think the speaker's statement does hold in the particular type of cases that he is talking about, and that the maps in the diagrams he speaks of are closed cofibrations, but that there are extra conditions on the diagrams which he is not mentioning. I would like any further insight that anyone can offer (even in general, about diagrams whose colimits and hocolimits are homotopy equivalent).
algebraic-topology category-theory homotopy-theory polyhedral-product
asked Dec 6 '18 at 2:47
MattMatt
2,577719
$endgroup$
$begingroup$
This is not true. Certain special cases hold. Can you include some more context or a timestamp?
$endgroup$
– Kevin Carlson
Dec 6 '18 at 3:59
$begingroup$
Ah, excuse me. I had meant to include the timestamp. It's at just after the 5 minute mark. I will edit this in to the question.
$endgroup$
– Matt
Dec 6 '18 at 7:20
1
$begingroup$
P.s. if anyone knows how to do that thing where the link to a youtube video actually takes you to a specific time in the video, I would like my link to do that, but I don't know how.
$endgroup$
– Matt
Dec 6 '18 at 9:02
3
$begingroup$
The statement the speaker is using is for filtered homotopy colimits. To add a timestamp, add &t=5m5s to the end of the link, or whatever specific minute/second mark you actually want.
$endgroup$
– Mike Miller
Dec 6 '18 at 11:50
1
$begingroup$
Right, the homotopy colimit of that diagram is a point. My counterexample was rather $*rightrightarrows *$.
$endgroup$
– Kevin Carlson
Dec 18 '18 at 16:18
|
show 5 more comments
$begingroup$
In this video, at just after the 5 minute mark, the speaker says:
"...the colimit of the diagram has the homotopy type of the homotopy colimit. Why? Because all of the maps included in the diagram are closed cofibrations, and there is a theorem that if you have a diagram where everything is a closed cofibration, the homotopy type of the colimit and the homotopy colimit are the same."
What is the theorem being referenced here, or alternatively how does this follow?
EDIT:
If I have the following diagram:
$$* rightrightarrows *$$
I think that this is a diagram of closed cofibrations for which the statement does not hold, for example.
I think the speaker's statement does hold in the particular type of cases that he is talking about, and that the maps in the diagrams he speaks of are closed cofibrations, but that there are extra conditions on the diagrams which he is not mentioning. I would like any further insight that anyone can offer (even in general, about diagrams whose colimits and hocolimits are homotopy equivalent).
algebraic-topology category-theory homotopy-theory polyhedral-product
asked Dec 6 '18 at 2:47
MattMatt
2,577719
$endgroup$
In this video, at just after the 5 minute mark, the speaker says:
"...the colimit of the diagram has the homotopy type of the homotopy colimit. Why? Because all of the maps included in the diagram are closed cofibrations, and there is a theorem that if you have a diagram where everything is a closed cofibration, the homotopy type of the colimit and the homotopy colimit are the same."
What is the theorem being referenced here, or alternatively how does this follow?
EDIT:
If I have the following diagram:
$$* rightrightarrows *$$
I think that this is a diagram of closed cofibrations for which the statement does not hold, for example.
I think the speaker's statement does hold in the particular type of cases that he is talking about, and that the maps in the diagrams he speaks of are closed cofibrations, but that there are extra conditions on the diagrams which he is not mentioning. I would like any further insight that anyone can offer (even in general, about diagrams whose colimits and hocolimits are homotopy equivalent).
algebraic-topology category-theory homotopy-theory polyhedral-product
algebraic-topology category-theory homotopy-theory polyhedral-product
asked Dec 6 '18 at 2:47
MattMatt
2,577719
asked Dec 6 '18 at 2:47
MattMatt
2,577719
edited Dec 18 '18 at 17:05
Matt
asked Dec 6 '18 at 2:47
MattMatt
2,577719
asked Dec 6 '18 at 2:47
MattMatt
2,577719
asked Dec 6 '18 at 2:47
MattMatt
2,577719
2,577719
$begingroup$
This is not true. Certain special cases hold. Can you include some more context or a timestamp?
$endgroup$
– Kevin Carlson
Dec 6 '18 at 3:59
$begingroup$
Ah, excuse me. I had meant to include the timestamp. It's at just after the 5 minute mark. I will edit this in to the question.
$endgroup$
– Matt
Dec 6 '18 at 7:20
1
$begingroup$
P.s. if anyone knows how to do that thing where the link to a youtube video actually takes you to a specific time in the video, I would like my link to do that, but I don't know how.
$endgroup$
– Matt
Dec 6 '18 at 9:02
3
$begingroup$
The statement the speaker is using is for filtered homotopy colimits. To add a timestamp, add &t=5m5s to the end of the link, or whatever specific minute/second mark you actually want.
$endgroup$
– Mike Miller
Dec 6 '18 at 11:50
1
$begingroup$
Right, the homotopy colimit of that diagram is a point. My counterexample was rather $*rightrightarrows *$.
$endgroup$
– Kevin Carlson
Dec 18 '18 at 16:18
|
show 5 more comments
$begingroup$
This is not true. Certain special cases hold. Can you include some more context or a timestamp?
$endgroup$
– Kevin Carlson
Dec 6 '18 at 3:59
$begingroup$
Ah, excuse me. I had meant to include the timestamp. It's at just after the 5 minute mark. I will edit this in to the question.
$endgroup$
– Matt
Dec 6 '18 at 7:20
1
$begingroup$
P.s. if anyone knows how to do that thing where the link to a youtube video actually takes you to a specific time in the video, I would like my link to do that, but I don't know how.
$endgroup$
– Matt
Dec 6 '18 at 9:02
3
$begingroup$
The statement the speaker is using is for filtered homotopy colimits. To add a timestamp, add &t=5m5s to the end of the link, or whatever specific minute/second mark you actually want.
$endgroup$
– Mike Miller
Dec 6 '18 at 11:50
1
$begingroup$
Right, the homotopy colimit of that diagram is a point. My counterexample was rather $*rightrightarrows *$.
$endgroup$
– Kevin Carlson
Dec 18 '18 at 16:18
$begingroup$
This is not true. Certain special cases hold. Can you include some more context or a timestamp?
$endgroup$
– Kevin Carlson
Dec 6 '18 at 3:59
$begingroup$
This is not true. Certain special cases hold. Can you include some more context or a timestamp?
$endgroup$
– Kevin Carlson
Dec 6 '18 at 3:59
$begingroup$
Ah, excuse me. I had meant to include the timestamp. It's at just after the 5 minute mark. I will edit this in to the question.
$endgroup$
– Matt
Dec 6 '18 at 7:20
$begingroup$
Ah, excuse me. I had meant to include the timestamp. It's at just after the 5 minute mark. I will edit this in to the question.
$endgroup$
– Matt
Dec 6 '18 at 7:20
1
1
$begingroup$
P.s. if anyone knows how to do that thing where the link to a youtube video actually takes you to a specific time in the video, I would like my link to do that, but I don't know how.
$endgroup$
– Matt
Dec 6 '18 at 9:02
$begingroup$
P.s. if anyone knows how to do that thing where the link to a youtube video actually takes you to a specific time in the video, I would like my link to do that, but I don't know how.
$endgroup$
– Matt
Dec 6 '18 at 9:02
3
3
$begingroup$
The statement the speaker is using is for filtered homotopy colimits. To add a timestamp, add &t=5m5s to the end of the link, or whatever specific minute/second mark you actually want.
$endgroup$
– Mike Miller
Dec 6 '18 at 11:50
$begingroup$
The statement the speaker is using is for filtered homotopy colimits. To add a timestamp, add &t=5m5s to the end of the link, or whatever specific minute/second mark you actually want.
$endgroup$
– Mike Miller
Dec 6 '18 at 11:50
1
1
$begingroup$
Right, the homotopy colimit of that diagram is a point. My counterexample was rather $*rightrightarrows *$.
$endgroup$
– Kevin Carlson
Dec 18 '18 at 16:18
$begingroup$
Right, the homotopy colimit of that diagram is a point. My counterexample was rather $*rightrightarrows *$.
$endgroup$
– Kevin Carlson
Dec 18 '18 at 16:18
|
show 5 more comments
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$begingroup$
This is not true. Certain special cases hold. Can you include some more context or a timestamp?
$endgroup$
– Kevin Carlson
Dec 6 '18 at 3:59
$begingroup$
Ah, excuse me. I had meant to include the timestamp. It's at just after the 5 minute mark. I will edit this in to the question.
$endgroup$
– Matt
Dec 6 '18 at 7:20
1
$begingroup$
P.s. if anyone knows how to do that thing where the link to a youtube video actually takes you to a specific time in the video, I would like my link to do that, but I don't know how.
$endgroup$
– Matt
Dec 6 '18 at 9:02
3
$begingroup$
The statement the speaker is using is for filtered homotopy colimits. To add a timestamp, add &t=5m5s to the end of the link, or whatever specific minute/second mark you actually want.
$endgroup$
– Mike Miller
Dec 6 '18 at 11:50
1
$begingroup$
Right, the homotopy colimit of that diagram is a point. My counterexample was rather $*rightrightarrows *$.
$endgroup$
– Kevin Carlson
Dec 18 '18 at 16:18