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I created minimal fibration which could be merged with minimal Kan fibration. The idea-section says that this notion is needed to give a well defined notion of n-category. However there are other applications which I didn’t mention.
Maybe your new page should be called minimal Joyal fibration for clarity?
BTW, on the nLab we generally use the terminology “$(\infty,1)$-category” and “$(n,1)$-category” where Lurie says “$\infty$-category” and “$n$-category”.
Yes, I made the $(-,1)$ corrections.
I don’t know if “minimal” in the sense of being homotopically determined by the value on the boundary doesn’t make sense for some other kind of fibration Joyal introduced, too. We could call the page minimal inner fibration?
Wait, changing “$n$-category” to “$(n,1)$-category” in the present context makes the problem worse, not better: the entry is about a specific model, not about the general notion.
There is the general notion of an $(\infty,1)$-category being: an object in $(\infty,1)Cat$. And then there are several specific presentations of $(\infty,1)Cat$ and fixing one of these gives us a particular model of $(\infty,1)$-categories. Quasi-categories are one such model. (Other models are: simplicial categories or categories with weak equivalences, or derivators, etc.)
The entry is about a property of a special model not of the general notion. Without fixing the particular presentation of $(\infty.1)Cat$ given by quasicategories/the Joyal model structure, it does not make sense to ask “Is a morphism a minimal fibration?”
Secondly, even with this understood, the term “$n$-category” is dangerous. I know that it is in Def. 2.3.4.1 in HTT, but it is nevertheless bad (and collides with use of (n,1)-topos later in the book). Because, again, that Def. 2.3.4.1 is about a property of a model for an $(n,1)$-category, not of an $(n,1)$-category as such.
What that definition 2.3.4.1 really defines is a quasi-categorical analog of “(n+1)-coskeletalness”. Namely it asks a quasi-category not only to present an $(n,1)$-category, but to do so in the strictest possible ways: namely so that not there are not only no “homotopy groups” above degree $n$, but so that there are actually no nontrivial cells above degree $n$.
I have edited the entry accordingly. Check out the new Idea-section!
I also changed the title of the entry, as Mike suggested. And as soon as I have faught the dreaded cache-bug, I will create a disambiguation entry titled minimal fibration that will point to the various sub-notion. There are also minimal dendroidal fibrations, for instance.
If “minimal inner fibration” is correct, then that would be a better name. I assumed that the proper analogue of Kan fibrations was Joyal fibrations (meaning fibrations in the quasi-category model structure), but I guess that these minimal fibrations are really just minimal inner fibrations.
True. I have renamed it once more. (My clearcache-script now running hot ;-)
I went and touched a bunch of related entries, such as fibration and fibrations of quasi-categories and so forth. Despite some serious effort with these entries, now and previously, it remains a bit of a mess. Not entirely our fault, of course.
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