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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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


















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











$endgroup$












  • $begingroup$
    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
















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











$endgroup$




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















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 11 at 8:58









Francesco Polizzi

48.2k3128210




48.2k3128210










asked Mar 11 at 8:40









Frank KongFrank Kong

385




385












  • $begingroup$
    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






$begingroup$
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












1 Answer
1






active

oldest

votes


















5












$begingroup$

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
    Mar 11 at 11:22










  • $begingroup$
    Thanks for your help!
    $endgroup$
    – Frank Kong
    Mar 11 at 12:04











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1 Answer
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active

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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









5












$begingroup$

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
    Mar 11 at 11:22










  • $begingroup$
    Thanks for your help!
    $endgroup$
    – Frank Kong
    Mar 11 at 12:04
















5












$begingroup$

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
    Mar 11 at 11:22










  • $begingroup$
    Thanks for your help!
    $endgroup$
    – Frank Kong
    Mar 11 at 12:04














5












5








5





$begingroup$

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 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














  • 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


















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