Definition A.3.1.5 of Higher Topos Theory












3












$begingroup$


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







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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
    10 hours ago


















3












$begingroup$


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







share|cite|improve this question









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
    10 hours ago
















3












3








3





$begingroup$


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







share|cite|improve this question









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






share|cite|improve this question









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.











share|cite|improve this question









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.









share|cite|improve this question




share|cite|improve this question








edited 10 hours ago









Francesco Polizzi

48.1k3128209




48.1k3128209






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 10 hours ago









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
    10 hours ago




















  • $begingroup$
    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
    10 hours ago


















$begingroup$
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
10 hours ago






$begingroup$
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
10 hours ago












1 Answer
1






active

oldest

votes


















4












$begingroup$

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.






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
    $endgroup$
    – Theo Johnson-Freyd
    8 hours ago










  • $begingroup$
    Thanks for your help!
    $endgroup$
    – Frank Kong
    7 hours ago











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4












$begingroup$

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.






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
    $endgroup$
    – Theo Johnson-Freyd
    8 hours ago










  • $begingroup$
    Thanks for your help!
    $endgroup$
    – Frank Kong
    7 hours ago
















4












$begingroup$

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.






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
    $endgroup$
    – Theo Johnson-Freyd
    8 hours ago










  • $begingroup$
    Thanks for your help!
    $endgroup$
    – Frank Kong
    7 hours ago














4












4








4





$begingroup$

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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 10 hours ago









Stefano AriottaStefano Ariotta

32148




32148








  • 1




    $begingroup$
    Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
    $endgroup$
    – Theo Johnson-Freyd
    8 hours ago










  • $begingroup$
    Thanks for your help!
    $endgroup$
    – Frank Kong
    7 hours ago














  • 1




    $begingroup$
    Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
    $endgroup$
    – Theo Johnson-Freyd
    8 hours ago










  • $begingroup$
    Thanks for your help!
    $endgroup$
    – Frank Kong
    7 hours ago








1




1




$begingroup$
Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
$endgroup$
– Theo Johnson-Freyd
8 hours ago




$begingroup$
Now I'm worried about Mike Shulman's question mathoverflow.net/questions/322917.
$endgroup$
– Theo Johnson-Freyd
8 hours ago












$begingroup$
Thanks for your help!
$endgroup$
– Frank Kong
7 hours ago




$begingroup$
Thanks for your help!
$endgroup$
– Frank Kong
7 hours ago










Frank Kong is a new contributor. Be nice, and check out our Code of Conduct.










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