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the entry fibrations of quasi-categories was getting too long for my taste. I have to change my original plans about it.
Now I split off left Kan fibration from it, which currently duplicates material from this entry and from fibration fibered in groupoids. I'll see how to eventualy harmonize this a bit better.
Presently my next immediate goal is to write out as a pedagogical introduction to the notion of left/right fibration a nice detailed proof for the fact that a functor is an op-fibration fibered in groupoids precisely if its nerve is a left Kan fibration.
I wanted to do that today, but got distracted. Now I am getting too tired. So I'll maybe postpone this until tomorrow...
Hey, Urs, I've got a question for you. Is there any chance you could write up some motivation for the "covariant model structure" in Lurie's HTT 2.1.4 (this is the title of the section)? I'm looking at it, and I don't understand what he's trying to do.
Is there any chance you could write up some motivation for the "covariant model structure"
have you looked at model structure for left fibrations? That's what it is: a model for the (oo,1)-category of (oo,1)-categories fibered in oo-groupoids over a fixed (oo,1)-category.
You can do me a favor by looking at this (stubby) page and telling me which further points you'd like to see discussed more.
Oh wow, you already wrote something up! You should add a redirect there from "covariant model structure" or something similar.
The only suggestion I have for that page is "connecting the motivation to the formalism". Typically, you can do this by showing how this reduces in the case of ordinary categories and also maybe giving a worked diagram or two showing "how everything fits together".
You should add a redirect there from "covariant model structure"
I don't want to use that term at all as an entry title or redirect. It may be okay in the context of a book, but on the nLab we have to try to find titles that are reasonably universally descriptive of the content of the entry, such as to avoid that future entries want to have a title that is already in use for a different notion.
And "covariant model structure" just is way too unspecific for that purpose. I want to name pages on model structures either after the categories that are equipped with the model structure, or -- probably preferably -- by the things that are modeled by the model structure.
And at left fibration and (infinity,1)-Grothendieck construction I announce the model structure for left fibrations explicitiy. I have added the term "covariant model structure" now in the References-section of that latter page. So eventually Google will make the connection and present the desired search results.
"connecting the motivation to the formalism"
Yes, I'll be busy expanding the details for a bit now...
Anyway, thanks! That makes a lot more sense now. I had tried looking up "covariant model structure" in the past, but I was unable to find it.
here now a Motivation section that aims to explain why the weak equivalences in the model structure are what they are
here the beginn ing of a section Examples for joins of quasicategories that introduces the joins that we need in the definition of these pushouts
Yeah, I was familiar with those (cones). Thanks a lot for all of the help!
sure, I know that you knew this. This was just a general announcement of stuff that I added.
finally finished typing the proof that under the nerve fibrations in groupoids are precisely right Kan fibrations
listed a bunch more properties of left anodyne morphisms in the last section at left fibration
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