Definition A.3.1.5 of Higher Topos TheorySimplicially enriched cartesian closed categoriesFibered/cofibered...
Definition A.3.1.5 of Higher Topos Theory
Simplicially enriched cartesian closed categoriesFibered/cofibered higher categories, relative model structures, slicing, and (∞,2)-category theoryWhat are the higher morphisms between enriched higher categories?Equivalences of Internal CategoriesThe state of the art in the rectification of homotopy-coherent structuresA model category of abelian categories?Is the theory of weak $n$-categories a cofibrant replacement of the theory of strict ones?The category theory of Span-enriched categories / 2-Segal spacesExcellent monoidal model categories admit enriched fibrant replacement functors?About a canonical model structure on topologically enriched categoriesTheorem 2.1.2.2 Higher Topos Theory
$begingroup$
I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition
of a $mathbf{S}$-enriched model category, where $mathbf{S}$ is a monoidal model category. But in the book model structures are introduced only on $mathsf{Set}$-enriched categories.
So, what does a model structure on a $mathbf{S}$-enriched category mean?
Is it supposed to be that $mathbf{S}$ obtains a forgetful functor to $mathsf{Set}$, and the model structure is defined on the category with respect to the $mathsf{Set}$ enrichment, or is it something else?
higher-category-theory model-categories
edited Mar 11 at 8:58
Francesco Polizzi
48.2k3128210
asked Mar 11 at 8:40
Frank KongFrank Kong
385
New contributor
Frank Kong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
$begingroup$
I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition
of a $mathbf{S}$-enriched model category, where $mathbf{S}$ is a monoidal model category. But in the book model structures are introduced only on $mathsf{Set}$-enriched categories.
So, what does a model structure on a $mathbf{S}$-enriched category mean?
Is it supposed to be that $mathbf{S}$ obtains a forgetful functor to $mathsf{Set}$, and the model structure is defined on the category with respect to the $mathsf{Set}$ enrichment, or is it something else?
higher-category-theory model-categories
edited Mar 11 at 8:58
Francesco Polizzi
48.2k3128210
asked Mar 11 at 8:40
Frank KongFrank Kong
385
New contributor
Frank Kong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
I suspect Lurie is using implicitly the lax monoidal forgetful functor $mathrm{Hom}_{mathbf{S}}(1_{mathbf{S}},-):mathbf{S}→mathrm{Set}$, although I'll confess I never really thought about it (but this gives the sensible choice of forgetful functor for both simplicial sets and chain complexes).
$endgroup$
– Denis Nardin
Mar 11 at 9:09
add a comment |
$begingroup$
I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition
of a $mathbf{S}$-enriched model category, where $mathbf{S}$ is a monoidal model category. But in the book model structures are introduced only on $mathsf{Set}$-enriched categories.
So, what does a model structure on a $mathbf{S}$-enriched category mean?
Is it supposed to be that $mathbf{S}$ obtains a forgetful functor to $mathsf{Set}$, and the model structure is defined on the category with respect to the $mathsf{Set}$ enrichment, or is it something else?
higher-category-theory model-categories
edited Mar 11 at 8:58
Francesco Polizzi
48.2k3128210
asked Mar 11 at 8:40
Frank KongFrank Kong
385
New contributor
Frank Kong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition
of a $mathbf{S}$-enriched model category, where $mathbf{S}$ is a monoidal model category. But in the book model structures are introduced only on $mathsf{Set}$-enriched categories.
So, what does a model structure on a $mathbf{S}$-enriched category mean?
Is it supposed to be that $mathbf{S}$ obtains a forgetful functor to $mathsf{Set}$, and the model structure is defined on the category with respect to the $mathsf{Set}$ enrichment, or is it something else?
higher-category-theory model-categories
higher-category-theory model-categories
edited Mar 11 at 8:58
Francesco Polizzi
48.2k3128210
asked Mar 11 at 8:40
Frank KongFrank Kong
385
New contributor
Frank Kong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Mar 11 at 8:58
Francesco Polizzi
48.2k3128210
asked Mar 11 at 8:40
Frank KongFrank Kong
385
New contributor
Frank Kong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Mar 11 at 8:58
Francesco Polizzi
48.2k3128210
edited Mar 11 at 8:58
Francesco Polizzi
48.2k3128210
edited Mar 11 at 8:58
Francesco Polizzi
48.2k3128210
48.2k3128210
asked Mar 11 at 8:40
Frank KongFrank Kong
385
New contributor
Frank Kong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Mar 11 at 8:40
Frank KongFrank Kong
385
asked Mar 11 at 8:40
Frank KongFrank Kong
385
385
New contributor
Frank Kong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Frank Kong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Frank Kong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
I suspect Lurie is using implicitly the lax monoidal forgetful functor $mathrm{Hom}_{mathbf{S}}(1_{mathbf{S}},-):mathbf{S}→mathrm{Set}$, although I'll confess I never really thought about it (but this gives the sensible choice of forgetful functor for both simplicial sets and chain complexes).
$endgroup$
– Denis Nardin
Mar 11 at 9:09
add a comment |
$begingroup$
I suspect Lurie is using implicitly the lax monoidal forgetful functor $mathrm{Hom}_{mathbf{S}}(1_{mathbf{S}},-):mathbf{S}→mathrm{Set}$, although I'll confess I never really thought about it (but this gives the sensible choice of forgetful functor for both simplicial sets and chain complexes).
$endgroup$
– Denis Nardin
Mar 11 at 9:09
$begingroup$
I suspect Lurie is using implicitly the lax monoidal forgetful functor $mathrm{Hom}_{mathbf{S}}(1_{mathbf{S}},-):mathbf{S}→mathrm{Set}$, although I'll confess I never really thought about it (but this gives the sensible choice of forgetful functor for both simplicial sets and chain complexes).
$endgroup$
– Denis Nardin
Mar 11 at 9:09
$begingroup$
I suspect Lurie is using implicitly the lax monoidal forgetful functor $mathrm{Hom}_{mathbf{S}}(1_{mathbf{S}},-):mathbf{S}→mathrm{Set}$, although I'll confess I never really thought about it (but this gives the sensible choice of forgetful functor for both simplicial sets and chain complexes).
$endgroup$
– Denis Nardin
Mar 11 at 9:09
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your guess is correct, indeed. In general, given any monoidal category $(mathbf V, otimes, 1)$, and any $mathbf V$-enriched category $mathbf C$, one can always consider the underlying category $mathbf C_0$ as the ($mathbf{Set}$-)category having as objects the same objects as $mathbf C$, and as hom-sets
$$
mathbf C_0(x,y):= mathbf V(1,mathbf{Hom}_{mathbf C}(x,y))
$$
You can easily work out how to define composition, after checking that $mathbf V(1,-)$ is a lax monoidal functor, as pointed out by Denis in the comments.
Now, if $mathbf S$ is a monoidal model category, and $mathbf A$ is a $mathbf S$-enriched category, "equipping $mathbf A$ with a model structure" just means "equipping $mathbf A_0$ with a model structure", whereas to talk about a $mathbf S$-enriched model category one requires the two extra conditions spelled out in HTT A.3.1.5.
answered Mar 11 at 9:15
Stefano AriottaStefano Ariotta
33148
$endgroup$
1
$begingroup$
Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
$endgroup$
– Theo Johnson-Freyd
Mar 11 at 11:22
$begingroup$
Thanks for your help!
$endgroup$
– Frank Kong
Mar 11 at 12:04
add a comment |
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$begingroup$
Your guess is correct, indeed. In general, given any monoidal category $(mathbf V, otimes, 1)$, and any $mathbf V$-enriched category $mathbf C$, one can always consider the underlying category $mathbf C_0$ as the ($mathbf{Set}$-)category having as objects the same objects as $mathbf C$, and as hom-sets
$$
mathbf C_0(x,y):= mathbf V(1,mathbf{Hom}_{mathbf C}(x,y))
$$
You can easily work out how to define composition, after checking that $mathbf V(1,-)$ is a lax monoidal functor, as pointed out by Denis in the comments.
Now, if $mathbf S$ is a monoidal model category, and $mathbf A$ is a $mathbf S$-enriched category, "equipping $mathbf A$ with a model structure" just means "equipping $mathbf A_0$ with a model structure", whereas to talk about a $mathbf S$-enriched model category one requires the two extra conditions spelled out in HTT A.3.1.5.
answered Mar 11 at 9:15
Stefano AriottaStefano Ariotta
33148
$endgroup$
1
$begingroup$
Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
$endgroup$
– Theo Johnson-Freyd
Mar 11 at 11:22
$begingroup$
Thanks for your help!
$endgroup$
– Frank Kong
Mar 11 at 12:04
add a comment |
$begingroup$
Your guess is correct, indeed. In general, given any monoidal category $(mathbf V, otimes, 1)$, and any $mathbf V$-enriched category $mathbf C$, one can always consider the underlying category $mathbf C_0$ as the ($mathbf{Set}$-)category having as objects the same objects as $mathbf C$, and as hom-sets
$$
mathbf C_0(x,y):= mathbf V(1,mathbf{Hom}_{mathbf C}(x,y))
$$
You can easily work out how to define composition, after checking that $mathbf V(1,-)$ is a lax monoidal functor, as pointed out by Denis in the comments.
Now, if $mathbf S$ is a monoidal model category, and $mathbf A$ is a $mathbf S$-enriched category, "equipping $mathbf A$ with a model structure" just means "equipping $mathbf A_0$ with a model structure", whereas to talk about a $mathbf S$-enriched model category one requires the two extra conditions spelled out in HTT A.3.1.5.
answered Mar 11 at 9:15
Stefano AriottaStefano Ariotta
33148
$endgroup$
1
$begingroup$
Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
$endgroup$
– Theo Johnson-Freyd
Mar 11 at 11:22
$begingroup$
Thanks for your help!
$endgroup$
– Frank Kong
Mar 11 at 12:04
add a comment |
$begingroup$
Your guess is correct, indeed. In general, given any monoidal category $(mathbf V, otimes, 1)$, and any $mathbf V$-enriched category $mathbf C$, one can always consider the underlying category $mathbf C_0$ as the ($mathbf{Set}$-)category having as objects the same objects as $mathbf C$, and as hom-sets
$$
mathbf C_0(x,y):= mathbf V(1,mathbf{Hom}_{mathbf C}(x,y))
$$
You can easily work out how to define composition, after checking that $mathbf V(1,-)$ is a lax monoidal functor, as pointed out by Denis in the comments.
Now, if $mathbf S$ is a monoidal model category, and $mathbf A$ is a $mathbf S$-enriched category, "equipping $mathbf A$ with a model structure" just means "equipping $mathbf A_0$ with a model structure", whereas to talk about a $mathbf S$-enriched model category one requires the two extra conditions spelled out in HTT A.3.1.5.
answered Mar 11 at 9:15
Stefano AriottaStefano Ariotta
33148
$endgroup$
Your guess is correct, indeed. In general, given any monoidal category $(mathbf V, otimes, 1)$, and any $mathbf V$-enriched category $mathbf C$, one can always consider the underlying category $mathbf C_0$ as the ($mathbf{Set}$-)category having as objects the same objects as $mathbf C$, and as hom-sets
$$
mathbf C_0(x,y):= mathbf V(1,mathbf{Hom}_{mathbf C}(x,y))
$$
You can easily work out how to define composition, after checking that $mathbf V(1,-)$ is a lax monoidal functor, as pointed out by Denis in the comments.
Now, if $mathbf S$ is a monoidal model category, and $mathbf A$ is a $mathbf S$-enriched category, "equipping $mathbf A$ with a model structure" just means "equipping $mathbf A_0$ with a model structure", whereas to talk about a $mathbf S$-enriched model category one requires the two extra conditions spelled out in HTT A.3.1.5.
answered Mar 11 at 9:15
Stefano AriottaStefano Ariotta
33148
answered Mar 11 at 9:15
Stefano AriottaStefano Ariotta
33148
answered Mar 11 at 9:15
Stefano AriottaStefano Ariotta
33148
answered Mar 11 at 9:15
Stefano AriottaStefano Ariotta
33148
33148
1
$begingroup$
Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
$endgroup$
– Theo Johnson-Freyd
Mar 11 at 11:22
$begingroup$
Thanks for your help!
$endgroup$
– Frank Kong
Mar 11 at 12:04
add a comment |
1
$begingroup$
Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
$endgroup$
– Theo Johnson-Freyd
Mar 11 at 11:22
$begingroup$
Thanks for your help!
$endgroup$
– Frank Kong
Mar 11 at 12:04
1
1
$begingroup$
Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
$endgroup$
– Theo Johnson-Freyd
Mar 11 at 11:22
$begingroup$
Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
$endgroup$
– Theo Johnson-Freyd
Mar 11 at 11:22
$begingroup$
Thanks for your help!
$endgroup$
– Frank Kong
Mar 11 at 12:04
$begingroup$
Thanks for your help!
$endgroup$
– Frank Kong
Mar 11 at 12:04
add a comment |
Frank Kong is a new contributor. Be nice, and check out our Code of Conduct.
Frank Kong is a new contributor. Be nice, and check out our Code of Conduct.
Frank Kong is a new contributor. Be nice, and check out our Code of Conduct.
Frank Kong is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
I suspect Lurie is using implicitly the lax monoidal forgetful functor $mathrm{Hom}_{mathbf{S}}(1_{mathbf{S}},-):mathbf{S}→mathrm{Set}$, although I'll confess I never really thought about it (but this gives the sensible choice of forgetful functor for both simplicial sets and chain complexes).
$endgroup$
– Denis Nardin
Mar 11 at 9:09