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edited Mar 14 at 22:53

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In a series of papers ([1], [2] and [3]), Dwyer and Kan introduced the hammock localization [2] as an effective technique to compute the simplicial localization of a model category [1]. This is meant to encode the higher homotopy implicit in a model category; indeed, if one takes $\pi_0$ hom-wise the result is the homotopy category of the original model category.

They say their main application is to reconstruct the function complexes between fibrant cofibrant objects in a simplicial model category. And they managed. Cool! On the other hand, a feature-oriented description of such a localization is given in [3]. Unfortunately, at that time (I guess!) the Lurie technology was not available, so a universal-property localization was not so easy to state in such a context. But now we do have it, so I wonder:

Given a model category $C$ with weak equivalences $\mathcal{W}$, consider its hammock localization $L^H(C, \mathcal{W})$. Take its derived homotopy coherent nerve $\textbf{R}N_{coh}(L^H(C,\mathcal{W}))$ and you will get a quasicategory. On the other hand, you can take the ordinary nerve of $C$, and then invert $\mathcal{W}$ in the sense of Lurie ([6], 1.3.4.1), obtaining $N(C)[\mathcal{W}^{-1}]$. Do these two constructions agree?

As a recall,Recall that the derived homotopy coherent nerve is obtained by first taking a fibrant replacement (akaa.k.a., locally Kan simplicial category) and then applying the homotopy coherent nerve (which will automatically be fibrant in $\textrm{sSets}$, i.e., a quasicategory).

The furthest I could get. There is a nice paper by Hinich in which some of these issues are discussed. In particular, if $C$ is a fibrant simplicial (say model) category with weak equivalences $\mathcal{W}$, there is a weak equivalence of marked simplicial sets (proposition 1.2.1): $$ (**) \ \ \ (N_{coh}C, \mathcal{W}) \to \textbf{R}N(L^H(C,\mathcal{W}))^{\natural} $$ Notice also that the right-hand side is fibrant in the category of marked simplicial sets (over the point): indeed, from HTT 3.1.4.1 one can easily deduce that fibrant here means beaing a quasicategory with equivalences as marked edges. Also, in remark 1.3.4.2. from HA, we see that $N_{coh}C[\mathcal{W}^{-1}]$ can be identified with a fibrant replacement of $(N_{coh}C,\mathcal{W})$ in marked simplicial sets. Since equation (**) gives an explicit fibrant replacement, we get the thesis.

Unluckily, proposition 1.2.1 from Hinich is available only in the case in which $C$ is a (fibrant) simplicial category, though it's not clear where the simplicial structure is used. In fact, hammock localization is designed (also) to provide a simplicial structure on an arbitrary model category when such an extra structure does not come for free. Thanks

Thanks for your help! It's not really my field and I find it hard sometimes to orient in the choose-your-model yoga.

Bibliography

[1] - Simplicial localization of categories, Dwyer and Kan
[2] - Calculating simplicial localization, Dwyer and Kan
[3] - Function Complexes in Homotopical Algebra, Dwyer and Kan
[4] - DK localization revisited, Vladimir Hinich
[5] - Higher Topos Theory, Jacob Lurie
[6] - Higher Algebra, Jacob Lurie

In a series of papers ([1], [2] and [3]), Dwyer and Kan introduced the hammock localization [2] as an effective technique to compute the simplicial localization of a model category [1]. This is meant to encode the higher homotopy implicit in a model category; indeed, if one takes $\pi_0$ hom-wise the result is the homotopy category of the original model category.

They say their main application is to reconstruct the function complexes between fibrant cofibrant objects in a simplicial model category. And they managed. Cool! On the other hand, a feature-oriented description of such a localization is given in [3]. Unfortunately, at that time (I guess!) the Lurie technology was not available, so a universal-property localization was not so easy to state in such a context. But now we do have it, so I wonder:

Given a model category $C$ with weak equivalences $\mathcal{W}$, consider its hammock localization $L^H(C, \mathcal{W})$. Take its derived homotopy coherent nerve $\textbf{R}N_{coh}(L^H(C,\mathcal{W}))$ and you will get a quasicategory. On the other hand, you can take the ordinary nerve of $C$, and then invert $\mathcal{W}$ in the sense of Lurie ([6], 1.3.4.1), obtaining $N(C)[\mathcal{W}^{-1}]$. Do these two constructions agree?

As a recall, the derived homotopy coherent nerve is obtained by first taking a fibrant replacement (aka locally Kan simplicial category) and then applying the homotopy coherent nerve (which will automatically be fibrant in $\textrm{sSets}$, i.e. a quasicategory).

The furthest I could get. There is a nice paper by Hinich in which some of these issues are discussed. In particular, if $C$ is a fibrant simplicial (say model) category with weak equivalences $\mathcal{W}$, there is a weak equivalence of marked simplicial sets (proposition 1.2.1): $$ (**) \ \ \ (N_{coh}C, \mathcal{W}) \to \textbf{R}N(L^H(C,\mathcal{W}))^{\natural} $$ Notice also that the right-hand side is fibrant in the category of marked simplicial sets (over the point): indeed, from HTT 3.1.4.1 one can easily deduce that fibrant here means beaing a quasicategory with equivalences as marked edges. Also, in remark 1.3.4.2. from HA, we see that $N_{coh}C[\mathcal{W}^{-1}]$ can be identified with a fibrant replacement of $(N_{coh}C,\mathcal{W})$ in marked simplicial sets. Since equation (**) gives an explicit fibrant replacement, we get the thesis.

Unluckily, proposition 1.2.1 from Hinich is available only in the case in which $C$ is a (fibrant) simplicial category, though it's not clear where the simplicial structure is used. In fact, hammock localization is designed (also) to provide a simplicial structure on an arbitrary model category when such an extra structure does not come for free. Thanks for your help! It's not really my field and I find it hard sometimes to orient in the choose-your-model yoga.

Bibliography

[1] - Simplicial localization of categories, Dwyer and Kan
[2] - Calculating simplicial localization, Dwyer and Kan
[3] - Function Complexes in Homotopical Algebra, Dwyer and Kan
[4] - DK localization revisited, Vladimir Hinich
[5] - Higher Topos Theory, Jacob Lurie
[6] - Higher Algebra, Jacob Lurie

In a series of papers ([1], [2] and [3]), Dwyer and Kan introduced the hammock localization [2] as an effective technique to compute the simplicial localization of a model category [1]. This is meant to encode the higher homotopy implicit in a model category; indeed, if one takes $\pi_0$ hom-wise the result is the homotopy category of the original model category.

They say their main application is to reconstruct the function complexes between fibrant cofibrant objects in a simplicial model category. And they managed. Cool! On the other hand, a feature-oriented description of such a localization is given in [3]. Unfortunately, at that time (I guess!) the Lurie technology was not available, so a universal-property localization was not so easy to state in such a context. But now we do have it, so I wonder:

Given a model category $C$ with weak equivalences $\mathcal{W}$, consider its hammock localization $L^H(C, \mathcal{W})$. Take its derived homotopy coherent nerve $\textbf{R}N_{coh}(L^H(C,\mathcal{W}))$ and you will get a quasicategory. On the other hand, you can take the ordinary nerve of $C$, and then invert $\mathcal{W}$ in the sense of Lurie ([6], 1.3.4.1), obtaining $N(C)[\mathcal{W}^{-1}]$. Do these two constructions agree?

Recall that the derived homotopy coherent nerve is obtained by first taking a fibrant replacement (a.k.a., locally Kan simplicial category) and then applying the homotopy coherent nerve (which will automatically be fibrant in $\textrm{sSets}$, i.e., a quasicategory).

The furthest I could get. There is a nice paper by Hinich in which some of these issues are discussed. In particular, if $C$ is a fibrant simplicial (say model) category with weak equivalences $\mathcal{W}$, there is a weak equivalence of marked simplicial sets (proposition 1.2.1): $$ (**) \ \ \ (N_{coh}C, \mathcal{W}) \to \textbf{R}N(L^H(C,\mathcal{W}))^{\natural} $$ Notice also that the right-hand side is fibrant in the category of marked simplicial sets (over the point): indeed, from HTT 3.1.4.1 one can easily deduce that fibrant here means beaing a quasicategory with equivalences as marked edges. Also, in remark 1.3.4.2. from HA, we see that $N_{coh}C[\mathcal{W}^{-1}]$ can be identified with a fibrant replacement of $(N_{coh}C,\mathcal{W})$ in marked simplicial sets. Since equation (**) gives an explicit fibrant replacement, we get the thesis.

Unluckily, proposition 1.2.1 from Hinich is available only in the case in which $C$ is a (fibrant) simplicial category, though it's not clear where the simplicial structure is used. In fact, hammock localization is designed (also) to provide a simplicial structure on an arbitrary model category when such an extra structure does not come for free.

Thanks for your help! It's not really my field and I find it hard sometimes to orient in the choose-your-model yoga.

Bibliography

[1] - Simplicial localization of categories, Dwyer and Kan
[2] - Calculating simplicial localization, Dwyer and Kan
[3] - Function Complexes in Homotopical Algebra, Dwyer and Kan
[4] - DK localization revisited, Vladimir Hinich
[5] - Higher Topos Theory, Jacob Lurie
[6] - Higher Algebra, Jacob Lurie

Hammock was not a person's name, so should not be capitalized.

Source Link

edited Jun 27, 2021 at 8:40

David White

30.3k
9
153
250

Is Hammockhammock localization a localization in the sense of Lurie?

In a series of papers ([1][1], [2] and [3]), Dwyer and Kan introduced the Hammockhammock localization [2] as an effective technique to compute the simplicial localization of a model category [1 1]. This is meant to encode the higher homotopy implicit in a model category; indeed, if one takes $\pi_0$ hom-wise the result is the homotopy category of the original model category.

They say their main application is to recontructreconstruct the function complexes between fibrant cofibrant objects in a simplicial model category. And they managed. Cool! On the other hand, a feature-oriented description of such a localization is given in [3]. Unfortunately, at that time (I guess!) the Lurie technology was not available, so a universal-property localization was not so easy to state in such a context. But now we do have it, so I wonder:

Given a model category $C$ with weak equivalences $\mathcal{W}$, consider its Hammock Localizationhammock localization $L^H(C, \mathcal{W})$. Take its derived homotopy coherent nerve $\textbf{R}N_{coh}(L^H(C,\mathcal{W}))$ and you will get a quasicategory. On the other hand, you can take the ordinary nerve of $C$, and then invert $\mathcal{W}$ in the sense of Lurie ([6], 1.3.4.1), obtaining $N(C)[\mathcal{W}^{-1}]$. Do these two constructions agree?

As a recall, the derived homotopy coherent nerve is obtained by first taking a fibrant replacement (aka locally kanKan simplicial category) and then applying the homotopy coherent nerve (which will automatically be fibrant in $\textrm{sSets}$, i.e. a quasicategory).

The furthest I could get. There is a nice paper by Hinich in which some of these issues are discussed. In particular, if $C$ is a fibrant simplicial (say model) category with weak equivalences $\mathcal{W}$, there is a weak equivalence of marked simplicial sets (proposition 1.2.1): $$ (**) \ \ \ (N_{coh}C, \mathcal{W}) \to \textbf{R}N(L^H(C,\mathcal{W}))^{\natural} $$ Notice also that the right-hand side is fibrant in the category of marked simplicial sets (over the point): indeed, from HTT 3.1.4.1 one can easily deduce that fibrant here means beaing a quasicategory with equivalences as marked edges. Also, in remark 1.3.4.2. from HA, we see that $N_{coh}C[\mathcal{W}^{-1}]$ can be identified with a fibrant replacement of $(N_{coh}C,\mathcal{W})$ in marked simplicial sets. Since equation (**) gives an explicit fibrant replacement, we get the thesis.

Unluckily, proposition 1.2.1 from Hinich is available only in the case in which $C$ is a (fibrant) simplicial category, though it's not clear where the simplicial structure is used. In fact, hammock localization is designed (also) to provide a simplicial structure on an arbitrary model category when such an extra structure does not come for free. Thanks for your help! It's not really my field and I find it hard sometimes to orient in the choose-your-model yoga.

Bibliography

[1 1] - Simplicial localization of categories, Dwyer and Kan
[2] - Calculating simplicial localization, Dwyer and Kan
[3] - Function Complexes in Homotopical Algebra, Dwyer and Kan
[4] - DK localization revisited, Vladimir Hinich
[5] - Higher Topos Theory, Jacob Lurie
[6] - Higher Algebra, Jacob Lurie

Is Hammock localization a localization in the sense of Lurie?

In a series of papers ([1], [2] and [3]), Dwyer and Kan introduced the Hammock localization [2] as an effective technique to compute the simplicial localization of a model category [1]. This is meant to encode the higher homotopy implicit in a model category; indeed, if one takes $\pi_0$ hom-wise the result is the homotopy category of the original model category.

They say their main application is to recontruct the function complexes between fibrant cofibrant objects in a simplicial model category. And they managed. Cool! On the other hand, a feature-oriented description of such a localization is given in [3]. Unfortunately, at that time (I guess!) the Lurie technology was not available, so a universal-property localization was not so easy to state in such a context. But now we do have it, so I wonder:

Given a model category $C$ with weak equivalences $\mathcal{W}$, consider its Hammock Localization $L^H(C, \mathcal{W})$. Take its derived homotopy coherent nerve $\textbf{R}N_{coh}(L^H(C,\mathcal{W}))$ and you will get a quasicategory. On the other hand, you can take the ordinary nerve of $C$, and then invert $\mathcal{W}$ in the sense of Lurie ([6], 1.3.4.1), obtaining $N(C)[\mathcal{W}^{-1}]$. Do these two constructions agree?

As a recall, the derived homotopy coherent nerve is obtained by first taking a fibrant replacement (aka locally kan simplicial category) and then applying the homotopy coherent nerve (which will automatically be fibrant in $\textrm{sSets}$, i.e. a quasicategory).

The furthest I could get. There is a nice paper by Hinich in which some of these issues are discussed. In particular, if $C$ is a fibrant simplicial (say model) category with weak equivalences $\mathcal{W}$, there is a weak equivalence of marked simplicial sets (proposition 1.2.1): $$ (**) \ \ \ (N_{coh}C, \mathcal{W}) \to \textbf{R}N(L^H(C,\mathcal{W}))^{\natural} $$ Notice also that the right-hand side is fibrant in the category of marked simplicial sets (over the point): indeed, from HTT 3.1.4.1 one can easily deduce that fibrant here means beaing a quasicategory with equivalences as marked edges. Also, in remark 1.3.4.2. from HA, we see that $N_{coh}C[\mathcal{W}^{-1}]$ can be identified with a fibrant replacement of $(N_{coh}C,\mathcal{W})$ in marked simplicial sets. Since equation (**) gives an explicit fibrant replacement, we get the thesis.

Unluckily, proposition 1.2.1 from Hinich is available only in the case in which $C$ is a (fibrant) simplicial category, though it's not clear where the simplicial structure is used. In fact, hammock localization is designed (also) to provide a simplicial structure on an arbitrary model category when such an extra structure does not come for free. Thanks for your help! It's not really my field and I find it hard sometimes to orient in the choose-your-model yoga.

Bibliography

[1] - Simplicial localization of categories, Dwyer and Kan
[2] - Calculating simplicial localization, Dwyer and Kan
[3] - Function Complexes in Homotopical Algebra, Dwyer and Kan
[4] - DK localization revisited, Vladimir Hinich
[5] - Higher Topos Theory, Jacob Lurie
[6] - Higher Algebra, Jacob Lurie

Is hammock localization a localization in the sense of Lurie?

In a series of papers ([1], [2] and [3]), Dwyer and Kan introduced the hammock localization [2] as an effective technique to compute the simplicial localization of a model category [1]. This is meant to encode the higher homotopy implicit in a model category; indeed, if one takes $\pi_0$ hom-wise the result is the homotopy category of the original model category.

They say their main application is to reconstruct the function complexes between fibrant cofibrant objects in a simplicial model category. And they managed. Cool! On the other hand, a feature-oriented description of such a localization is given in [3]. Unfortunately, at that time (I guess!) the Lurie technology was not available, so a universal-property localization was not so easy to state in such a context. But now we do have it, so I wonder:

Given a model category $C$ with weak equivalences $\mathcal{W}$, consider its hammock localization $L^H(C, \mathcal{W})$. Take its derived homotopy coherent nerve $\textbf{R}N_{coh}(L^H(C,\mathcal{W}))$ and you will get a quasicategory. On the other hand, you can take the ordinary nerve of $C$, and then invert $\mathcal{W}$ in the sense of Lurie ([6], 1.3.4.1), obtaining $N(C)[\mathcal{W}^{-1}]$. Do these two constructions agree?

As a recall, the derived homotopy coherent nerve is obtained by first taking a fibrant replacement (aka locally Kan simplicial category) and then applying the homotopy coherent nerve (which will automatically be fibrant in $\textrm{sSets}$, i.e. a quasicategory).

The furthest I could get. There is a nice paper by Hinich in which some of these issues are discussed. In particular, if $C$ is a fibrant simplicial (say model) category with weak equivalences $\mathcal{W}$, there is a weak equivalence of marked simplicial sets (proposition 1.2.1): $$ (**) \ \ \ (N_{coh}C, \mathcal{W}) \to \textbf{R}N(L^H(C,\mathcal{W}))^{\natural} $$ Notice also that the right-hand side is fibrant in the category of marked simplicial sets (over the point): indeed, from HTT 3.1.4.1 one can easily deduce that fibrant here means beaing a quasicategory with equivalences as marked edges. Also, in remark 1.3.4.2. from HA, we see that $N_{coh}C[\mathcal{W}^{-1}]$ can be identified with a fibrant replacement of $(N_{coh}C,\mathcal{W})$ in marked simplicial sets. Since equation (**) gives an explicit fibrant replacement, we get the thesis.

Unluckily, proposition 1.2.1 from Hinich is available only in the case in which $C$ is a (fibrant) simplicial category, though it's not clear where the simplicial structure is used. In fact, hammock localization is designed (also) to provide a simplicial structure on an arbitrary model category when such an extra structure does not come for free. Thanks for your help! It's not really my field and I find it hard sometimes to orient in the choose-your-model yoga.

Bibliography