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created homotopy Kan extension
this is to go along with the discussion at limit in a quasi-category that I am currently working on.
I feel like in this context, the name Kan extension is just confusing. That is, unless it has a nice little homotopy-theoretic definition. Yes, I know you didn't make up the name. I am just making a remark =).
the name Kan extension is just confusing.
Why do you think that? This gives literally the global definition of Kan extension. The name is compelling.
I added a section that briefly recollects the ordinary notion of Kan extension, to be generalized here.
I'm saying that in the context of simplicial sets, kan anything that isnt' related to the model structure is misleading. The ordinary Kan extension has nothing to do with a model structure. (Or maybe it does? Kan lifts and Kan extensions look like they might be related to some kind of crazy model structure, but I don't know).
Not only Kan anything but also anything anything in 1-categories is usually not related to model structures. But if you look for (infty,1)-categorifications they may be realized via model category presentations. So the categorifications are model related. Isn't that the case here ?
Presumably it doesn't have to do with the Kan model structure, though.
Of course not. Why would anybody make such a strange restriction for a universal-type construction ?
Yes, that is my point. You've got this thing called a Kan extension when you're talking about simplicial sets which have a thing called the kan model structure. I dunno, it just seems like one of those things that would be misleading to someone new is all.
You've got this thing called a Kan extension
Look Kan extensions have their meaning in category theory for decades without any reference to anything simplicial. Why would that suddenly change in infinity-categorification ? I expect that a newcomer to infinity categories knows reasonable amount of 1-category theory before hand. Otherwise he will be quickly lost.
Harry,
it is unfortunate that Dan Kan had had more than one good idea. But it so happens that Kan extension (have a look!) is a bed-rock term in category theory. Are you familiar with it?
And it's not the Kan model structure, but the Quillen model structure on simplicial sets.
Yes, I'm familiar with kan extensions. I was making an inconsequential remark. It doesn't really matter, I was half-kidding.
it is unfortunate that Dan Kan had had more than one good idea.
May we all be so lucky as to have enough good ideas that our name is not exclusively associated with one of them by posterity. (-:
SOSHWIS ;)
Why are you wasting time on hours long intended joke discussions ? There is so much you could help in other entries...
I'm procrastinating to avoid doing homework, obviously.
and you expect some support ? :)
spelled out the full proof that derived hom into a homotopy Kan extension is a homotopy Kan extension
expanded the section on homotopy Kan extensions in Kan-complex enriched catetgories, one of the two crucial ingredients in showing that model-category theoretic homotopy Kan extensions do model quasi-categorical Kan extensions (at least for lims and colims.)
added some basic citations on pointwise homotopy Kan extensions at homotopy Kan extension – Properties – Pointwise
It was pointed out on MO that the page homotopy Kan extension claimed that the domain categories $C,C'$ were also simplicial model categories, when in fact what was meant was that they were just simplicially enriched categories. I fixed it.
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