Model category of all model categories
4
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Is there some natural model category structure on the category of all small model categories and some specified functors among them (i.e. not necessarily all functors), say Quillen adjunctions? What is the most natural choice to this question: what should the classses of fibrations, cofibrations and weak equivalencies be?
category-theory functors model-categories fibration cofibrations
asked Dec 1 '18 at 13:27
user122424user122424
1,1212716
$endgroup$
|
show 2 more comments
4
$begingroup$
Is there some natural model category structure on the category of all small model categories and some specified functors among them (i.e. not necessarily all functors), say Quillen adjunctions? What is the most natural choice to this question: what should the classses of fibrations, cofibrations and weak equivalencies be?
category-theory functors model-categories fibration cofibrations
asked Dec 1 '18 at 13:27
user122424user122424
1,1212716
$endgroup$
$begingroup$
To define a category, you have to be sure that for every object $X,Y$ $Hom(X,Y)$ is a set. Are you sure that the set of functors between two categories endowed witha Quillen model is a set ?
$endgroup$
– Tsemo Aristide
Dec 1 '18 at 15:41
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Well, we may encounter an illegitimate category $mathbf{CAT}$.
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– user122424
Dec 2 '18 at 16:03
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I've edited my question by inserting the word "small". Everything goes smoothly now ?
$endgroup$
– user122424
Dec 2 '18 at 16:42
$begingroup$
Well, very few model categories of interest are small...Anyway, that's not really the point. A model category structure gives, in particular, homotopy limits and colimits, and it's extremely difficult to construct even these for model categories. Julie Bergner is one author who's written papers on special cases like homotopy fiber products. Much easier is to study similar questions $infty$-categorically.
$endgroup$
– Kevin Carlson
Dec 2 '18 at 21:13
2
$begingroup$
@KevinCarlson "Very few model categories of interest are small". Maybe that's true in your universe. I'm in the next one up, and I see lots of interesting small model categories!
$endgroup$
– Alex Kruckman
Dec 3 '18 at 16:30
|
show 2 more comments
4
4
4
0
$begingroup$
Is there some natural model category structure on the category of all small model categories and some specified functors among them (i.e. not necessarily all functors), say Quillen adjunctions? What is the most natural choice to this question: what should the classses of fibrations, cofibrations and weak equivalencies be?
category-theory functors model-categories fibration cofibrations
asked Dec 1 '18 at 13:27
user122424user122424
1,1212716
$endgroup$
Is there some natural model category structure on the category of all small model categories and some specified functors among them (i.e. not necessarily all functors), say Quillen adjunctions? What is the most natural choice to this question: what should the classses of fibrations, cofibrations and weak equivalencies be?
category-theory functors model-categories fibration cofibrations
category-theory functors model-categories fibration cofibrations
asked Dec 1 '18 at 13:27
user122424user122424
1,1212716
asked Dec 1 '18 at 13:27
user122424user122424
1,1212716
edited Dec 2 '18 at 16:41
user122424
asked Dec 1 '18 at 13:27
user122424user122424
1,1212716
asked Dec 1 '18 at 13:27
user122424user122424
1,1212716
asked Dec 1 '18 at 13:27
user122424user122424
1,1212716
1,1212716
$begingroup$
To define a category, you have to be sure that for every object $X,Y$ $Hom(X,Y)$ is a set. Are you sure that the set of functors between two categories endowed witha Quillen model is a set ?
$endgroup$
– Tsemo Aristide
Dec 1 '18 at 15:41
$begingroup$
Well, we may encounter an illegitimate category $mathbf{CAT}$.
$endgroup$
– user122424
Dec 2 '18 at 16:03
$begingroup$
I've edited my question by inserting the word "small". Everything goes smoothly now ?
$endgroup$
– user122424
Dec 2 '18 at 16:42
$begingroup$
Well, very few model categories of interest are small...Anyway, that's not really the point. A model category structure gives, in particular, homotopy limits and colimits, and it's extremely difficult to construct even these for model categories. Julie Bergner is one author who's written papers on special cases like homotopy fiber products. Much easier is to study similar questions $infty$-categorically.
$endgroup$
– Kevin Carlson
Dec 2 '18 at 21:13
2
$begingroup$
@KevinCarlson "Very few model categories of interest are small". Maybe that's true in your universe. I'm in the next one up, and I see lots of interesting small model categories!
$endgroup$
– Alex Kruckman
Dec 3 '18 at 16:30
|
show 2 more comments
$begingroup$
To define a category, you have to be sure that for every object $X,Y$ $Hom(X,Y)$ is a set. Are you sure that the set of functors between two categories endowed witha Quillen model is a set ?
$endgroup$
– Tsemo Aristide
Dec 1 '18 at 15:41
$begingroup$
Well, we may encounter an illegitimate category $mathbf{CAT}$.
$endgroup$
– user122424
Dec 2 '18 at 16:03
$begingroup$
I've edited my question by inserting the word "small". Everything goes smoothly now ?
$endgroup$
– user122424
Dec 2 '18 at 16:42
$begingroup$
Well, very few model categories of interest are small...Anyway, that's not really the point. A model category structure gives, in particular, homotopy limits and colimits, and it's extremely difficult to construct even these for model categories. Julie Bergner is one author who's written papers on special cases like homotopy fiber products. Much easier is to study similar questions $infty$-categorically.
$endgroup$
– Kevin Carlson
Dec 2 '18 at 21:13
2
$begingroup$
@KevinCarlson "Very few model categories of interest are small". Maybe that's true in your universe. I'm in the next one up, and I see lots of interesting small model categories!
$endgroup$
– Alex Kruckman
Dec 3 '18 at 16:30
$begingroup$
To define a category, you have to be sure that for every object $X,Y$ $Hom(X,Y)$ is a set. Are you sure that the set of functors between two categories endowed witha Quillen model is a set ?
$endgroup$
– Tsemo Aristide
Dec 1 '18 at 15:41
$begingroup$
To define a category, you have to be sure that for every object $X,Y$ $Hom(X,Y)$ is a set. Are you sure that the set of functors between two categories endowed witha Quillen model is a set ?
$endgroup$
– Tsemo Aristide
Dec 1 '18 at 15:41
$begingroup$
Well, we may encounter an illegitimate category $mathbf{CAT}$.
$endgroup$
– user122424
Dec 2 '18 at 16:03
$begingroup$
Well, we may encounter an illegitimate category $mathbf{CAT}$.
$endgroup$
– user122424
Dec 2 '18 at 16:03
$begingroup$
I've edited my question by inserting the word "small". Everything goes smoothly now ?
$endgroup$
– user122424
Dec 2 '18 at 16:42
$begingroup$
I've edited my question by inserting the word "small". Everything goes smoothly now ?
$endgroup$
– user122424
Dec 2 '18 at 16:42
$begingroup$
Well, very few model categories of interest are small...Anyway, that's not really the point. A model category structure gives, in particular, homotopy limits and colimits, and it's extremely difficult to construct even these for model categories. Julie Bergner is one author who's written papers on special cases like homotopy fiber products. Much easier is to study similar questions $infty$-categorically.
$endgroup$
– Kevin Carlson
Dec 2 '18 at 21:13
$begingroup$
Well, very few model categories of interest are small...Anyway, that's not really the point. A model category structure gives, in particular, homotopy limits and colimits, and it's extremely difficult to construct even these for model categories. Julie Bergner is one author who's written papers on special cases like homotopy fiber products. Much easier is to study similar questions $infty$-categorically.
$endgroup$
– Kevin Carlson
Dec 2 '18 at 21:13
2
2
$begingroup$
@KevinCarlson "Very few model categories of interest are small". Maybe that's true in your universe. I'm in the next one up, and I see lots of interesting small model categories!
$endgroup$
– Alex Kruckman
Dec 3 '18 at 16:30
$begingroup$
@KevinCarlson "Very few model categories of interest are small". Maybe that's true in your universe. I'm in the next one up, and I see lots of interesting small model categories!
$endgroup$
– Alex Kruckman
Dec 3 '18 at 16:30
|
show 2 more comments
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$begingroup$
To define a category, you have to be sure that for every object $X,Y$ $Hom(X,Y)$ is a set. Are you sure that the set of functors between two categories endowed witha Quillen model is a set ?
$endgroup$
– Tsemo Aristide
Dec 1 '18 at 15:41
$begingroup$
Well, we may encounter an illegitimate category $mathbf{CAT}$.
$endgroup$
– user122424
Dec 2 '18 at 16:03
$begingroup$
I've edited my question by inserting the word "small". Everything goes smoothly now ?
$endgroup$
– user122424
Dec 2 '18 at 16:42
$begingroup$
Well, very few model categories of interest are small...Anyway, that's not really the point. A model category structure gives, in particular, homotopy limits and colimits, and it's extremely difficult to construct even these for model categories. Julie Bergner is one author who's written papers on special cases like homotopy fiber products. Much easier is to study similar questions $infty$-categorically.
$endgroup$
– Kevin Carlson
Dec 2 '18 at 21:13
2
$begingroup$
@KevinCarlson "Very few model categories of interest are small". Maybe that's true in your universe. I'm in the next one up, and I see lots of interesting small model categories!
$endgroup$
– Alex Kruckman
Dec 3 '18 at 16:30