nForum - Discussion Feed (model structure for left fibrations) 2023-04-01T16:21:20+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher nLab edit announcer comments on "model structure for left fibrations" (83443) https://nforum.ncatlab.org/discussion/915/?Focus=83443#Comment_83443 2020-03-26T01:32:16+00:00 2023-04-01T16:21:19+00:00 nLab edit announcer https://nforum.ncatlab.org/account/1691/ In theorem 3.3 describing a set of maps to localize at, the left horn was mistakenly notated with a \Delta rather than a \Lambda. Anonymous diff, v18, current

In theorem 3.3 describing a set of maps to localize at, the left horn was mistakenly notated with a \Delta rather than a \Lambda.

Anonymous

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Zhen Lin comments on "model structure for left fibrations" (53849) https://nforum.ncatlab.org/discussion/915/?Focus=53849#Comment_53849 2015-07-07T05:23:10+00:00 2023-04-01T16:21:19+00:00 Zhen Lin https://nforum.ncatlab.org/account/318/ Sure. That’s also a formal consequence of the claim that the left fibration model structure is a left Bousfield localisation of the sliced Joyal model structure.

Sure. That’s also a formal consequence of the claim that the left fibration model structure is a left Bousfield localisation of the sliced Joyal model structure.

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Danny Stevenson comments on "model structure for left fibrations" (53847) https://nforum.ncatlab.org/discussion/915/?Focus=53847#Comment_53847 2015-07-07T01:11:13+00:00 2023-04-01T16:21:19+00:00 Danny Stevenson https://nforum.ncatlab.org/account/692/ Hi, sorry to trouble you; I guess also in #13 above it is being implicitly used that every left fibration is an isofibration? (Of course this is well known to be true, probably the most direct way ... Hi, sorry to trouble you; I guess also in #13 above it is being implicitly used that every left fibration is an isofibration? (Of course this is well known to be true, probably the most direct way to see this is to use the description of covariant equivalences in [HTT, Definition 2.1.4.5]). ]]> Mike Shulman comments on "model structure for left fibrations" (53838) https://nforum.ncatlab.org/discussion/915/?Focus=53838#Comment_53838 2015-07-05T20:58:44+00:00 2023-04-01T16:21:19+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Okay, fair enough. Although HTT 2.1.2.7 is already more than half the way towards Lurie’s construction of the left fibration model structure.

Okay, fair enough. Although HTT 2.1.2.7 is already more than half the way towards Lurie’s construction of the left fibration model structure.

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Zhen Lin comments on "model structure for left fibrations" (53836) https://nforum.ncatlab.org/discussion/915/?Focus=53836#Comment_53836 2015-07-05T07:25:30+00:00 2023-04-01T16:21:19+00:00 Zhen Lin https://nforum.ncatlab.org/account/318/ It’s obvious that &Sscr;\mathcal{S}-local in the Joyal-enriched sense implies &Sscr;\mathcal{S}-local in the ordinary sense (for fibrant objects).

It’s obvious that $\mathcal{S}$-local in the Joyal-enriched sense implies $\mathcal{S}$-local in the ordinary sense (for fibrant objects).

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Mike Shulman comments on "model structure for left fibrations" (53835) https://nforum.ncatlab.org/discussion/915/?Focus=53835#Comment_53835 2015-07-05T02:51:14+00:00 2023-04-01T16:21:19+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Every time I say “Bousfield localization”, I mean in the ordinary sense, not the Joyal-enriched sense.

Every time I say “Bousfield localization”, I mean in the ordinary sense, not the Joyal-enriched sense.

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Zhen Lin comments on "model structure for left fibrations" (53832) https://nforum.ncatlab.org/discussion/915/?Focus=53832#Comment_53832 2015-07-04T08:37:18+00:00 2023-04-01T16:21:19+00:00 Zhen Lin https://nforum.ncatlab.org/account/318/ Well, let’s see. We know every &Sscr;\mathcal{S}-local object is a left fibration. Conversely, an isofibration C&rightarrow;SC \to S is &Sscr;\mathcal{S}-local in the Joyal-enriched ...

Well, let’s see. We know every $\mathcal{S}$-local object is a left fibration. Conversely, an isofibration $C \to S$ is $\mathcal{S}$-local in the Joyal-enriched sense if and only if it has the right lifting property with respect to $\Delta^m \times \Lambda^n_0 \cup \partial \Delta^m \times \Delta^n \hookrightarrow \Delta^m \times \Delta^n$. But that inclusion is a left anodyne extension by [HTT, Corollary 2.1.2.7], so we are done.

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Mike Shulman comments on "model structure for left fibrations" (53831) https://nforum.ncatlab.org/discussion/915/?Focus=53831#Comment_53831 2015-07-04T06:33:09+00:00 2023-04-01T16:21:19+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ But can you show that the fibrant objects in the Bousfield localization are precisely the left fibrations without already knowing that the model structure as defined by Lurie exists?

But can you show that the fibrant objects in the Bousfield localization are precisely the left fibrations without already knowing that the model structure as defined by Lurie exists?

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Zhen Lin comments on "model structure for left fibrations" (53827) https://nforum.ncatlab.org/discussion/915/?Focus=53827#Comment_53827 2015-07-03T07:41:42+00:00 2023-04-01T16:21:19+00:00 Zhen Lin https://nforum.ncatlab.org/account/318/ Re #9: Yes, of course. One of the irritations of this abuse of notation. The definition of weak equivalence appearing on the page (and in HTT) is rather mysterious. (It is much too clever for a ...

Re #9: Yes, of course. One of the irritations of this abuse of notation.

The definition of weak equivalence appearing on the page (and in HTT) is rather mysterious. (It is much too clever for a definition!) I think I like this Bousfield localisation definition better.

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Mike Shulman comments on "model structure for left fibrations" (53826) https://nforum.ncatlab.org/discussion/915/?Focus=53826#Comment_53826 2015-07-03T03:43:09+00:00 2023-04-01T16:21:20+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ I have added a mention of this fact to the page model structure for left fibrations.

I have added a mention of this fact to the page model structure for left fibrations.

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Mike Shulman comments on "model structure for left fibrations" (53825) https://nforum.ncatlab.org/discussion/915/?Focus=53825#Comment_53825 2015-07-03T03:37:47+00:00 2023-04-01T16:21:20+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ By the way, I assumed you meant the set of all maps of the form &Lambda; 0 n&hookrightarrow;&Delta; n\Lambda^n_0 \hookrightarrow \Delta^n over XX, i.e. indexed not just by nn but by a map ...

By the way, I assumed you meant the set of all maps of the form $\Lambda^n_0 \hookrightarrow \Delta^n$ over $X$, i.e. indexed not just by $n$ but by a map $\Delta^n\to X$.

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Mike Shulman comments on "model structure for left fibrations" (53823) https://nforum.ncatlab.org/discussion/915/?Focus=53823#Comment_53823 2015-07-02T16:34:30+00:00 2023-04-01T16:21:20+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Okay. And I think your argument works for ordinary localization too, replacing the internal hom-objects with their maximal subgroupoids and “isofibration” with Kan fibration. Pedro Boavida has ...

Okay. And I think your argument works for ordinary localization too, replacing the internal hom-objects with their maximal subgroupoids and “isofibration” with Kan fibration.

Pedro Boavida has just pointed out to me that your $\mathcal{S}$ in #7 is also the answer asserted by Moerdijk-Heuts on page 5 here.

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Zhen Lin comments on "model structure for left fibrations" (53822) https://nforum.ncatlab.org/discussion/915/?Focus=53822#Comment_53822 2015-07-01T19:49:00+00:00 2023-04-01T16:21:20+00:00 Zhen Lin https://nforum.ncatlab.org/account/318/ Hmmm, maybe the &Sscr;\mathcal{S} I suggested doesn’t quite work. How about &Sscr;={&Lambda; 0 n&hookrightarrow;&Delta; n:n&gt;0}\mathcal{S} = \{ \Lambda^n_0 \hookrightarrow ...

Hmmm, maybe the $\mathcal{S}$ I suggested doesn’t quite work. How about $\mathcal{S} = \{ \Lambda^n_0 \hookrightarrow \Delta^n : n \gt 0 \}$ instead?

1. If $f : A \to B$ is a monomorphism and $C$ is an isofibration, then $f^* : \underline{Hom}(B, C) \to \underline{Hom}(A, C)$ is an isofibration.
2. So if $C$ is an isofibration and $\mathcal{S}$-local in the Joyal-enriched sense, then $C$ has the right lifting property with respect to all members of $\mathcal{S}$, hence is a left fibration.
3. If $C$ is a left fibration, then it is $\mathcal{S}$-local in the Joyal-enriched sense, because the members of $\mathcal{S}$ are weak equivalences in the left model structure.
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Mike Shulman comments on "model structure for left fibrations" (53821) https://nforum.ncatlab.org/discussion/915/?Focus=53821#Comment_53821 2015-07-01T15:45:37+00:00 2023-04-01T16:21:20+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Hmm, when you have an enriched model structure I think you actually get two different notions of locality depending on whether you use the enriched hom-objects or the model-categorical hom-spaces, ...

Hmm, when you have an enriched model structure I think you actually get two different notions of locality depending on whether you use the enriched hom-objects or the model-categorical hom-spaces, and the latter is the one that corresponds to the usual sort of Bousfield localization. So I don’t believe your second bullet; instead I would expect to see $f^*$ acting on maximal sub-Kan-complexes of those mapping spaces.

Secondly, how do you know that your $\mathcal{S}$ (which is the obvious choice) does in fact detect the left fibrations?

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Zhen Lin comments on "model structure for left fibrations" (53819) https://nforum.ncatlab.org/discussion/915/?Focus=53819#Comment_53819 2015-07-01T09:14:02+00:00 2023-04-01T16:21:20+00:00 Zhen Lin https://nforum.ncatlab.org/account/318/ Well, the left model structure on sSet /X\mathbf{sSet}_{/ X} is enriched with respect to the Kan–Quillen model structure on sSet\mathbf{sSet}, and all objects are cofibrant, so the following is ...

Well, the left model structure on $\mathbf{sSet}_{/ X}$ is enriched with respect to the Kan–Quillen model structure on $\mathbf{sSet}$, and all objects are cofibrant, so the following is true:

• A morphism $f : A \to B$ in $\mathbf{sSet}_{/ X}$ is a weak equivalence in the left model structure if and only if $f^* : \underline{Hom}(B, C) \to \underline{Hom}(A, C)$ is a (homotopy) equivalence of Kan complexes for every left fibration $C$ over $X$.

(I am abusing notation by omitting the projections to $X$.) On the other hand, given a set $\mathcal{S}$ of morphisms in $\mathbf{sSet}_{/ X}$, since the slice Joyal model structure is enriched with respect to the Joyal model structure on $\mathbf{sSet}$, the following should be true:

• An isofibration $C$ over $X$ is an $\mathcal{S}$-local object in $\mathbf{sSet}_{/ X}$ if and only if $f^* : \underline{Hom}(B, C) \to \underline{Hom}(A, C)$ is a (categorical) equivalence of quasicategories for every $f : A \to B$ in $\mathcal{S}$.

Putting these two together, it would seem to me that to get the desired $\mathcal{S}$, it is enough to check that the $\mathcal{S}$-local objects you get are the left fibrations. So I think $\mathcal{S} = \{ \{ 0 \} \hookrightarrow \Delta^n : n \ge 0 \}$ works.

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Mike Shulman comments on "model structure for left fibrations" (53814) https://nforum.ncatlab.org/discussion/915/?Focus=53814#Comment_53814 2015-06-30T19:08:35+00:00 2023-04-01T16:21:20+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Suppose XX is a quasicategory. The model structure for left fibrations on sSet/XsSet/X is a left Bousfield localization of the model structure on an over category arising from the model structure ...

Suppose $X$ is a quasicategory. The model structure for left fibrations on $sSet/X$ is a left Bousfield localization of the model structure on an over category arising from the model structure for quasi-categories, since it has the same cofibrations and fewer fibrant objects. Can one exhibit a nice small set of maps at which it is the localization?

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Urs comments on "model structure for left fibrations" (27878) https://nforum.ncatlab.org/discussion/915/?Focus=27878#Comment_27878 2011-12-18T23:30:16+00:00 2023-04-01T16:21:20+00:00 Urs https://nforum.ncatlab.org/account/4/ Thanks for catching this. I have fixed it. Not sure why the “left” got missing.

Thanks for catching this. I have fixed it. Not sure why the “left” got missing.

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Mike Shulman comments on "model structure for left fibrations" (27875) https://nforum.ncatlab.org/discussion/915/?Focus=27875#Comment_27875 2011-12-18T22:53:15+00:00 2023-04-01T16:21:20+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ The page model structure for left fibrations claims that this model structure is proper. However, the cited propositions in HTT only claim that it is left proper. Is it right proper? If so, where ...

The page model structure for left fibrations claims that this model structure is proper. However, the cited propositions in HTT only claim that it is left proper. Is it right proper? If so, where is a proof?

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Urs comments on "model structure for left fibrations" (6364) https://nforum.ncatlab.org/discussion/915/?Focus=6364#Comment_6364 2010-03-15T14:51:30+00:00 2023-04-01T16:21:20+00:00 Urs https://nforum.ncatlab.org/account/4/ a few more details at model structure for left fibrations

a few more details at model structure for left fibrations

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