Model category of all model categories
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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 at 13:27
user122424
1,0971616
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up vote
2
down vote
favorite
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 at 13:27
user122424
1,0971616
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 ?
– Tsemo Aristide
Dec 1 at 15:41
Well, we may encounter an illegitimate category $mathbf{CAT}$.
– user122424
2 days ago
I've edited my question by inserting the word "small". Everything goes smoothly now ?
– user122424
2 days ago
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.
– Kevin Carlson
2 days ago
1
@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!
– Alex Kruckman
yesterday
|
show 2 more comments
up vote
2
down vote
favorite
up vote
2
down vote
favorite
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 at 13:27
user122424
1,0971616
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 at 13:27
user122424
1,0971616
asked Dec 1 at 13:27
user122424
1,0971616
edited 2 days ago
asked Dec 1 at 13:27
user122424
1,0971616
asked Dec 1 at 13:27
user122424
1,0971616
asked Dec 1 at 13:27
user122424
1,0971616
1,0971616
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 ?
– Tsemo Aristide
Dec 1 at 15:41
Well, we may encounter an illegitimate category $mathbf{CAT}$.
– user122424
2 days ago
I've edited my question by inserting the word "small". Everything goes smoothly now ?
– user122424
2 days ago
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.
– Kevin Carlson
2 days ago
1
@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!
– Alex Kruckman
yesterday
|
show 2 more comments
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 ?
– Tsemo Aristide
Dec 1 at 15:41
Well, we may encounter an illegitimate category $mathbf{CAT}$.
– user122424
2 days ago
I've edited my question by inserting the word "small". Everything goes smoothly now ?
– user122424
2 days ago
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.
– Kevin Carlson
2 days ago
1
@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!
– Alex Kruckman
yesterday
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 ?
– Tsemo Aristide
Dec 1 at 15:41
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 ?
– Tsemo Aristide
Dec 1 at 15:41
Well, we may encounter an illegitimate category $mathbf{CAT}$.
– user122424
2 days ago
Well, we may encounter an illegitimate category $mathbf{CAT}$.
– user122424
2 days ago
I've edited my question by inserting the word "small". Everything goes smoothly now ?
– user122424
2 days ago
I've edited my question by inserting the word "small". Everything goes smoothly now ?
– user122424
2 days ago
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.
– Kevin Carlson
2 days ago
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.
– Kevin Carlson
2 days ago
1
1
@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!
– Alex Kruckman
yesterday
@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!
– Alex Kruckman
yesterday
|
show 2 more comments
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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 ?
– Tsemo Aristide
Dec 1 at 15:41
Well, we may encounter an illegitimate category $mathbf{CAT}$.
– user122424
2 days ago
I've edited my question by inserting the word "small". Everything goes smoothly now ?
– user122424
2 days ago
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.
– Kevin Carlson
2 days ago
1
@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!
– Alex Kruckman
yesterday