Theorem 2.1.2.2 Higher Topos Theory
At the page 74 of HTT, there is the following theorem
Let $S$ be a simplicial set, $mathcal{C}$ a simplicial category, and $phi: mathfrak{C}[S] rightarrow mathcal{C}^{op}$ a simplicial functor. The straightening and unstraigntening functors determine a Quillen adjunction
$$ St_{phi} : (Set_{Delta})_{/S} leftrightarrows Set_{Delta}^{mathcal{C}} :Un_{phi}$$
where $(Set_{Delta})_{/S}$ is endowed with the contravariant model structure and $Set_{Delta}^{mathcal{C}}$ with the projective model structure. [...]
In then says that the proof is easy, but I can't manage to show that $St_{phi}$ sends cofibrations to projective cofibrations. I thought that since the the class of morphisms which are sent to projective cofibrations is weakly saturated it is enough to show the result for all inclusions $partial Delta^n subseteq Delta^n$.
I did not have much success for the simplicial category $mathcal{C}$ and the map $phi$ could be anything and I have a hard time dealing with it.
Furthermore there is something else which troubles me: the model structure on the $Set^{mathcal{C}}_{Delta}$ makes no use of the simplicial enrichement on both $mathcal{C}$ and $sSet$ so I was wondering if I was not missing something by believing that the that model structure on $Set^{mathcal{C}}_{Delta}$ is really the projective model structure coming from the Kan model structure on $sSet$ and not the one coming somehow from an other model stucture using the simplicial enrichement.
category-theory simplicial-stuff higher-category-theory model-categories
edited Nov 25 '18 at 21:05
Arnaud D.
15.7k52343
asked Nov 25 '18 at 17:10
Oscar P.
494
add a comment |
At the page 74 of HTT, there is the following theorem
Let $S$ be a simplicial set, $mathcal{C}$ a simplicial category, and $phi: mathfrak{C}[S] rightarrow mathcal{C}^{op}$ a simplicial functor. The straightening and unstraigntening functors determine a Quillen adjunction
$$ St_{phi} : (Set_{Delta})_{/S} leftrightarrows Set_{Delta}^{mathcal{C}} :Un_{phi}$$
where $(Set_{Delta})_{/S}$ is endowed with the contravariant model structure and $Set_{Delta}^{mathcal{C}}$ with the projective model structure. [...]
In then says that the proof is easy, but I can't manage to show that $St_{phi}$ sends cofibrations to projective cofibrations. I thought that since the the class of morphisms which are sent to projective cofibrations is weakly saturated it is enough to show the result for all inclusions $partial Delta^n subseteq Delta^n$.
I did not have much success for the simplicial category $mathcal{C}$ and the map $phi$ could be anything and I have a hard time dealing with it.
Furthermore there is something else which troubles me: the model structure on the $Set^{mathcal{C}}_{Delta}$ makes no use of the simplicial enrichement on both $mathcal{C}$ and $sSet$ so I was wondering if I was not missing something by believing that the that model structure on $Set^{mathcal{C}}_{Delta}$ is really the projective model structure coming from the Kan model structure on $sSet$ and not the one coming somehow from an other model stucture using the simplicial enrichement.
category-theory simplicial-stuff higher-category-theory model-categories
edited Nov 25 '18 at 21:05
Arnaud D.
15.7k52343
asked Nov 25 '18 at 17:10
Oscar P.
494
2
I agree that it's not easy. Much of chapters 2-4 has been written in easier to read ways since HTT came out. I'd take a look at Moerdijk and Heuts for a more complete argument: arxiv.org/pdf/1308.0704.pdf
– Kevin Carlson
Nov 25 '18 at 17:36
Thank you for the link, I will take a look at it. However, Lurie seems to have a relatively simple argument in mind and I was really wondering what it could be.
– Oscar P.
Nov 26 '18 at 10:25
You might find someone to answer this at MO.
– Kevin Carlson
Nov 26 '18 at 18:59
Cross-posted: mathoverflow.net/questions/316393
– Watson
Nov 28 '18 at 10:37
add a comment |
At the page 74 of HTT, there is the following theorem
Let $S$ be a simplicial set, $mathcal{C}$ a simplicial category, and $phi: mathfrak{C}[S] rightarrow mathcal{C}^{op}$ a simplicial functor. The straightening and unstraigntening functors determine a Quillen adjunction
$$ St_{phi} : (Set_{Delta})_{/S} leftrightarrows Set_{Delta}^{mathcal{C}} :Un_{phi}$$
where $(Set_{Delta})_{/S}$ is endowed with the contravariant model structure and $Set_{Delta}^{mathcal{C}}$ with the projective model structure. [...]
In then says that the proof is easy, but I can't manage to show that $St_{phi}$ sends cofibrations to projective cofibrations. I thought that since the the class of morphisms which are sent to projective cofibrations is weakly saturated it is enough to show the result for all inclusions $partial Delta^n subseteq Delta^n$.
I did not have much success for the simplicial category $mathcal{C}$ and the map $phi$ could be anything and I have a hard time dealing with it.
Furthermore there is something else which troubles me: the model structure on the $Set^{mathcal{C}}_{Delta}$ makes no use of the simplicial enrichement on both $mathcal{C}$ and $sSet$ so I was wondering if I was not missing something by believing that the that model structure on $Set^{mathcal{C}}_{Delta}$ is really the projective model structure coming from the Kan model structure on $sSet$ and not the one coming somehow from an other model stucture using the simplicial enrichement.
category-theory simplicial-stuff higher-category-theory model-categories
edited Nov 25 '18 at 21:05
Arnaud D.
15.7k52343
asked Nov 25 '18 at 17:10
Oscar P.
494
At the page 74 of HTT, there is the following theorem
Let $S$ be a simplicial set, $mathcal{C}$ a simplicial category, and $phi: mathfrak{C}[S] rightarrow mathcal{C}^{op}$ a simplicial functor. The straightening and unstraigntening functors determine a Quillen adjunction
$$ St_{phi} : (Set_{Delta})_{/S} leftrightarrows Set_{Delta}^{mathcal{C}} :Un_{phi}$$
where $(Set_{Delta})_{/S}$ is endowed with the contravariant model structure and $Set_{Delta}^{mathcal{C}}$ with the projective model structure. [...]
In then says that the proof is easy, but I can't manage to show that $St_{phi}$ sends cofibrations to projective cofibrations. I thought that since the the class of morphisms which are sent to projective cofibrations is weakly saturated it is enough to show the result for all inclusions $partial Delta^n subseteq Delta^n$.
I did not have much success for the simplicial category $mathcal{C}$ and the map $phi$ could be anything and I have a hard time dealing with it.
Furthermore there is something else which troubles me: the model structure on the $Set^{mathcal{C}}_{Delta}$ makes no use of the simplicial enrichement on both $mathcal{C}$ and $sSet$ so I was wondering if I was not missing something by believing that the that model structure on $Set^{mathcal{C}}_{Delta}$ is really the projective model structure coming from the Kan model structure on $sSet$ and not the one coming somehow from an other model stucture using the simplicial enrichement.
category-theory simplicial-stuff higher-category-theory model-categories
category-theory simplicial-stuff higher-category-theory model-categories
edited Nov 25 '18 at 21:05
Arnaud D.
15.7k52343
asked Nov 25 '18 at 17:10
Oscar P.
494
edited Nov 25 '18 at 21:05
Arnaud D.
15.7k52343
asked Nov 25 '18 at 17:10
Oscar P.
494
edited Nov 25 '18 at 21:05
Arnaud D.
15.7k52343
edited Nov 25 '18 at 21:05
Arnaud D.
15.7k52343
edited Nov 25 '18 at 21:05
Arnaud D.
15.7k52343
15.7k52343
asked Nov 25 '18 at 17:10
Oscar P.
494
asked Nov 25 '18 at 17:10
Oscar P.
494
asked Nov 25 '18 at 17:10
Oscar P.
494
494
2
I agree that it's not easy. Much of chapters 2-4 has been written in easier to read ways since HTT came out. I'd take a look at Moerdijk and Heuts for a more complete argument: arxiv.org/pdf/1308.0704.pdf
– Kevin Carlson
Nov 25 '18 at 17:36
Thank you for the link, I will take a look at it. However, Lurie seems to have a relatively simple argument in mind and I was really wondering what it could be.
– Oscar P.
Nov 26 '18 at 10:25
You might find someone to answer this at MO.
– Kevin Carlson
Nov 26 '18 at 18:59
Cross-posted: mathoverflow.net/questions/316393
– Watson
Nov 28 '18 at 10:37
add a comment |
2
I agree that it's not easy. Much of chapters 2-4 has been written in easier to read ways since HTT came out. I'd take a look at Moerdijk and Heuts for a more complete argument: arxiv.org/pdf/1308.0704.pdf
– Kevin Carlson
Nov 25 '18 at 17:36
Thank you for the link, I will take a look at it. However, Lurie seems to have a relatively simple argument in mind and I was really wondering what it could be.
– Oscar P.
Nov 26 '18 at 10:25
You might find someone to answer this at MO.
– Kevin Carlson
Nov 26 '18 at 18:59
Cross-posted: mathoverflow.net/questions/316393
– Watson
Nov 28 '18 at 10:37
2
2
I agree that it's not easy. Much of chapters 2-4 has been written in easier to read ways since HTT came out. I'd take a look at Moerdijk and Heuts for a more complete argument: arxiv.org/pdf/1308.0704.pdf
– Kevin Carlson
Nov 25 '18 at 17:36
I agree that it's not easy. Much of chapters 2-4 has been written in easier to read ways since HTT came out. I'd take a look at Moerdijk and Heuts for a more complete argument: arxiv.org/pdf/1308.0704.pdf
– Kevin Carlson
Nov 25 '18 at 17:36
Thank you for the link, I will take a look at it. However, Lurie seems to have a relatively simple argument in mind and I was really wondering what it could be.
– Oscar P.
Nov 26 '18 at 10:25
Thank you for the link, I will take a look at it. However, Lurie seems to have a relatively simple argument in mind and I was really wondering what it could be.
– Oscar P.
Nov 26 '18 at 10:25
You might find someone to answer this at MO.
– Kevin Carlson
Nov 26 '18 at 18:59
You might find someone to answer this at MO.
– Kevin Carlson
Nov 26 '18 at 18:59
Cross-posted: mathoverflow.net/questions/316393
– Watson
Nov 28 '18 at 10:37
Cross-posted: mathoverflow.net/questions/316393
– Watson
Nov 28 '18 at 10:37
add a comment |
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I agree that it's not easy. Much of chapters 2-4 has been written in easier to read ways since HTT came out. I'd take a look at Moerdijk and Heuts for a more complete argument: arxiv.org/pdf/1308.0704.pdf
– Kevin Carlson
Nov 25 '18 at 17:36
Thank you for the link, I will take a look at it. However, Lurie seems to have a relatively simple argument in mind and I was really wondering what it could be.
– Oscar P.
Nov 26 '18 at 10:25
You might find someone to answer this at MO.
– Kevin Carlson
Nov 26 '18 at 18:59
Cross-posted: mathoverflow.net/questions/316393
– Watson
Nov 28 '18 at 10:37