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quick note local fibration
simplicial local fibration
While the literature mostly considers it for sheaves with values in $\mathrm{sSet}_{Quillen}$, there is no real need for that restriction. The notion of local fibration as such makes sense for sheaves with values in other model categories. I haven’t thought much about what happens to the standard statements under this generalization, but it would seem to me that by and large they ought to go through.
The page local fibration says that local fibrations
may instead form the fibrations in the corresponding category of fibrant objects-structure on simplicial sheaves
Where is a reference for that? I couldn’t find anything about it in a brief glance through the reference Local homotopy theory, but maybe I missed it. In particular, what is the meaning of “corresponding” — is this category of fibrant objects obtained in a canonical way from the model structures?
A standard – in fact the motivating – example of a category of fibrant objects is simplicial sheaves on a site with enough points, with fibrations and weak equivalences given stalkwise. It’s easy to check. The first reference that states it is BrownAHT.
Urs, thank you for pointing this out. Once you reminded me of BrownAHT, the whole thing makes so much more sense.
There is another meaning for local fibration (in $Top$), usually with adjectives attached (e.g. ’local Hurewicz|Serre fibration’), which is the exact analogue of ’local isomorphism’: a map $f:X\to Y$ is a local fibration if for every $x\in X$ there are open nhds $U$ of $x$ and $V$ of $f(x)$ such that the restriction $f|_U : U\to V$ is a fibration. This notion is used in the theory of topological stacks. My brain is not on at the moment, so is this encapsulable in Urs’ general definition in #3 in some general sense?
Thanks Urs. Can you say anything about what was meant by “corresponding”? Are local fibrations related to, or obtained from, the model structures in any way?
Mike writes:
Can you say anything about what was meant by “corresponding”?
I was in a haste when writing this and simply meant to allude to: given a site with enough points, the default structure of a category of fibrant objects on the locally fibrant simplicial sheaves.
Sorry for not having enough time for the entry. We were working on some article, wrote down the proof that pullbacks of local fibrations are homotopy pullbacks, came to wonder that this must have been noticed before, did some googling, and discovered the statement in that book by Jardine, wrote a brief $n$Lab entry such as not to forget the reference and had to run again to take care of a thousand other things.
I’ll try to expand the entry more in a while, maybe after we have our article out. It seems that with a bit of care one can tell a nice story here where one obtains a quite useful machinery by combined use of an injective/projective model structure on simplicial presheaves together with the “corresponding” structure of a category of fibrant objects on the locally fibrant simplicial sheaves. Each of these machineries is good for something at which the other is bad.
David writes:
a map $f:X\to Y$ is a local fibration if for every $x\in X$ there are open nhds $U$ of $x$ and $V$ of $f(x)$ such that the restriction $f|_U : U\to V$ is a fibration. This notion is used in the theory of topological stacks. My brain is not on at the moment, so is this encapsulable in Urs’ general definition in #3 in some general sense?
With a tad of manufacturing this should be a special case: regard $X \to Y$ as a morphism in $sSh(Y)$. Then the notions should coincide.
Jim writes:
A standard if fine but THE motivating!!
I can’t help it, but it has been the motivating example for Brown in his seminal article BrownAHT. It is what gives the article it’s very title. So it’s a historical fact that it has been the motivating example. I can’t help it. But, to be frank, I also don’t see what’s wrong with it. It’s a magnificent example.
Jim and Urs, you are both right, there is a misunderstanding. The entry is about a general notion, not only on AHT article. So you are both right: it is the moti vating example for Brown’s article as well as for a category of fibrant objects in the strict sense of his formal definition, but it is possibly only a motivating example for the kind of categories of fibrant objects in less formal sense, what probably Jim meant.
Urs, no need to apologize; thanks for clarifying. I was interested because I’m spending a lot of time thinking about univalence and categorical models of type theory these days. Injective model structures on simplicial presheaves have all the abstract nice properties that we want for interpreting type theory, but because their fibrations are so inexplicit, it’s difficult to see how you could build a univalent universe in them. So, I was wondering whether it might be possible to use local fibrations instead.
Thanks, Urs. I won’t sit down and do it now, but it’s helpful to know.
Zoran,
yes. Tonight in my sleep I realized what’s going on: Jim is thinking of what David is thinking of in #7 and so we are all talking past each other!
I will try to expand the entry and clarify. But not now.
Injective model structures on simplicial presheaves have all the abstract nice properties that we want for interpreting type theory, but because their fibrations are so inexplicit,
So why not use the projective model structure? What properties does that not have which you need? Of course for them it’s cofibrancy is hard, but in this case we have Dugger’s result on cofibrant replacement, so one can do much more.
Another by the way: categories of fibrant objects used to have the disadvantage that they were good for getting the homotopy category, but didn’t help with getting the simplicial localization. But a while back Thomas Nikolaus observed that the nerves of the evident cocycle categories in a cat of fibrant objects do indeed have the correct homotopy type of the derived hom space. This was going to be part of our upcoming article, only that we now realized that Denis-Charles Cisinski has already shown precisely this here. But in any case, it means that locally fibrant simplicial sheaves give a pretty comprehensive presentation of the oo-topos.
Not sure what result of Dugger’s you’re referring to. The problem with injective model structures is that the (acyclic) cofibrations are (probably?) not stable under pullback along fibrations, which is as far as we know necessary for interpreting identity types.
I mean the results in Universal homotopy theories:
A sufficient condition for a simplicial presheaf to be projectively cofibrant is for it to be degreewise a coproduct of representables with degenercies splitting off as a direct summand, see here;
there is an explicit functorial replacement by such objects, see here.
I found that most useful in applications. I can get my fibrant objects under control in the projective structure, and then this result of Dugger’s gives me good control over the cofibrant objects.
But of course it may not help for your purposes.
The problem with injective model structures is that the (acyclic) cofibrations are (probably?) not stable under pullback along fibrations,
Hm, aren’t they? Monos are pullback stable and the structure is right proper. (Easy to see if the site has enough points, but true generally, e.g. theorem 2 here.)
Sorry, I mistyped in a bad way; in #16 I meant the problem with the projective model structure, since that was what you were asking “why not use?”. That property does of course hold for the injective structure, which is one reason that that one is good.
Ah, I see. Hm, right, I don’t see why or if it should hold in the projective structure.
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