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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 8th 2010
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   Author: Harry Gindi
   Format: MarkdownItexI stated this for presheaf categories, but I'm pretty sure that it carries over for any Grothendieck topos. Check it out: [[lawvere interval]]

   I stated this for presheaf categories, but I’m pretty sure that it carries over for any Grothendieck topos.

   Check it out: lawvere interval

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 8th 2010
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   Author: Urs
   Format: MarkdownItexThanks. So is this the result of your quest to find a natural orighin of the model structure for Cartesian fibrations: you are claiming that its some construction by Cisisnki applied to the Lawvere cylinder in the preseaf topos of simplicial sets? I can't quite parse yet the details of your example at the end. I am sure I could if I now went to Cisinski's book and looked things up. But since I need to be doing something else, maybe you could expand a bit around where you say "we get exactly". By doing what exactly? When you go to the entry you'll see that I did some editorial editing. Feel free to change if you don't approve of what I did.

   Thanks.

   So is this the result of your quest to find a natural orighin of the model structure for Cartesian fibrations: you are claiming that its some construction by Cisisnki applied to the Lawvere cylinder in the preseaf topos of simplicial sets?

   I can’t quite parse yet the details of your example at the end. I am sure I could if I now went to Cisinski’s book and looked things up. But since I need to be doing something else, maybe you could expand a bit around where you say “we get exactly”. By doing what exactly?

   When you go to the entry you’ll see that I did some editorial editing. Feel free to change if you don’t approve of what I did.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 8th 2010
   • (edited Dec 8th 2010)
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   Author: Harry Gindi
   Format: MarkdownItex@Urs: I plan on finishing the article later. Basically, Cisinski has a construction that given a cylinder, a set of monomorphisms, and a set of generating cofibrations, we can complete the set of cofibrations to a class of anodyne morphisms for the cylinder. Then, by applying the machinery in chapter 1.3, we get a model structure. What I wrote up at the bottom there is from private correspondence with Cisinski (I asked him how to get Joyal's right anodyne morphisms using his formalism. He said that the way to do it is to generate the anodyne maps by taking Lawvere's cylinder and generating a class of anodyne morphisms using his construction). This is kinda easy to see, in some sense, since in particular, it generates the class of morphisms $\partial\Delta^n\times \Delta^1\coprod_{\partial \Delta^n\times \{1\}} \Delta^n\times \{1\}\to \Delta^n\times \Delta^1$, so it obviously contains the class of right anondyne morphisms, but the class of right anodyne morphisms basically has to contain all of the "extra" stuff lying around, so this is identical to the class of right anondyne morphisms. (Note that I'm claiming this is the contravariant model structure, not the cartesian model structure) Regarding the cartesian fibrations thing, I'm not trying it over simplicial sets, but over a different presheaf category where things are easier to encode (either that category of relative simplices or the category $\Delta^+$).

   @Urs: I plan on finishing the article later. Basically, Cisinski has a construction that given a cylinder, a set of monomorphisms, and a set of generating cofibrations, we can complete the set of cofibrations to a class of anodyne morphisms for the cylinder. Then, by applying the machinery in chapter 1.3, we get a model structure. What I wrote up at the bottom there is from private correspondence with Cisinski (I asked him how to get Joyal’s right anodyne morphisms using his formalism. He said that the way to do it is to generate the anodyne maps by taking Lawvere’s cylinder and generating a class of anodyne morphisms using his construction). This is kinda easy to see, in some sense, since in particular, it generates the class of morphisms $\partial\Delta^n\times \Delta^1\coprod_{\partial \Delta^n\times \{1\}} \Delta^n\times \{1\}\to \Delta^n\times \Delta^1$, so it obviously contains the class of right anondyne morphisms, but the class of right anodyne morphisms basically has to contain all of the “extra” stuff lying around, so this is identical to the class of right anondyne morphisms.

   (Note that I’m claiming this is the contravariant model structure, not the cartesian model structure)

   Regarding the cartesian fibrations thing, I’m not trying it over simplicial sets, but over a different presheaf category where things are easier to encode (either that category of relative simplices or the category $\Delta^+$).

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 8th 2010
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   Author: Urs
   Format: MarkdownItexOkay, I'd be quite interested in this, but can't look into it right now. But maybe later when you have found time to add mode details. > (Note that I'm claiming this is the contravariant model structure, not the cartesian model structure) Right, I changed the link to point to [[model structure for right fibrations]]. Let's use that term. Using terms like "contravariant model structure" may work in a book such as Lurie's where it does not collide with anything else, but here on the Lab we should use a more descriptive term.

   Okay, I’d be quite interested in this, but can’t look into it right now. But maybe later when you have found time to add mode details.

   (Note that I’m claiming this is the contravariant model structure, not the cartesian model structure)

   Right, I changed the link to point to model structure for right fibrations. Let’s use that term. Using terms like “contravariant model structure” may work in a book such as Lurie’s where it does not collide with anything else, but here on the Lab we should use a more descriptive term.

5. • CommentRowNumber5.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 8th 2010
   • (edited Dec 8th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexBy the way, something very interesting about Cisinski's construction that I didn't realize before: The "data" for his construction can be sliced in a way so they generate the model structure for right fibrations over any base, but if you slice the model structure, you get the wrong answer (for instance, slicing the model structure for right fibrations based at the terminal object, you get the relative Kan model structure, but if you slice the generator data over an object and complete to a model structure, you get the model structure for right fibrations over the slice. Also, the "wrong" slice is a left bousfield localization of the "correct" slice.

   By the way, something very interesting about Cisinski’s construction that I didn’t realize before: The “data” for his construction can be sliced in a way so they generate the model structure for right fibrations over any base, but if you slice the model structure, you get the wrong answer (for instance, slicing the model structure for right fibrations based at the terminal object, you get the relative Kan model structure, but if you slice the generator data over an object and complete to a model structure, you get the model structure for right fibrations over the slice.

   Also, the “wrong” slice is a left bousfield localization of the “correct” slice.

6. • CommentRowNumber6.
   • CommentAuthorjonsterling
   • CommentTimeMay 27th 2023
   • (edited May 27th 2023)
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   Author: jonsterling
   Format: MarkdownItexI noticed that this page claims that in an arbitrary presheaf topos, the terminal object has exactly two subobjects and that the subobject classifier has exactly two global points. This is obviously false (unless I am misreading?), but I wanted to understand what was intended here before I fixed it...

   I noticed that this page claims that in an arbitrary presheaf topos, the terminal object has exactly two subobjects and that the subobject classifier has exactly two global points. This is obviously false (unless I am misreading?), but I wanted to understand what was intended here before I fixed it…

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 27th 2023
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   Author: Todd_Trimble
   Format: MarkdownItexDoesn't look like you're misreading. I'm not sure what was intended. It was written a long time ago, but I think the author Harry Gindi is still active as a mathematician (Edinburgh, last I heard).

   Doesn’t look like you’re misreading. I’m not sure what was intended. It was written a long time ago, but I think the author Harry Gindi is still active as a mathematician (Edinburgh, last I heard).

8. • CommentRowNumber8.
   • CommentAuthorHurkyl
   • CommentTimeMay 27th 2023
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   Author: Hurkyl
   Format: MarkdownItexSince the original author was thinking about simplicial sets, maybe they had in mind a two-valued topos. Taking the bottom and top as endpoints is the correct thing to define the Lawvere interval; it's just the uniqueness statement that was in error.

   Since the original author was thinking about simplicial sets, maybe they had in mind a two-valued topos.

   Taking the bottom and top as endpoints is the correct thing to define the Lawvere interval; it’s just the uniqueness statement that was in error.

9. • CommentRowNumber9.
   • CommentAuthorjonsterling
   • CommentTimeMay 29th 2023
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   Author: jonsterling
   Format: MarkdownItexThanks! I've fixed it now.

   Thanks! I’ve fixed it now.