Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I added to basic localizer the statement that the notion is self-dual (even if the definition isn’t!) and a redirect from basic localizor (since that is the spelling Cisinski uses).
There also appears to be an incorrect assertion on the page:
For instance, the poset $\mathbb{N}$ has a contractible nerve, but its identity functor is not connected to a constant one by any natural zigzag.
But $\mathbb{N}$ has an initial object $0$, so there is in fact a one-step homotopy from the constant functor with value $0$ to $id$… no? Perhaps the intended example something like
$\bullet \leftarrow \bullet \rightarrow \bullet \leftarrow \bullet \rightarrow \bullet \leftarrow \bullet \rightarrow \cdots$which, as far as I can tell, does have a contractible nerve.
I have cross-linked basic localizer a bit, say with Cisinski model structure and with canonical model structure. Also expanded the bibliographical information and anchored it to receive links from the theorems.
But the entry badly needs some paragraph on what a basic localizer is actually good for, what it induces, and why!
@Zhen: you’re right, I think that must be the intended example. Feel free to fix it.
@Urs: Yes, that would be nice. I would like to read such a paragraph too. The only use I know of for the notion is Cisinski’s theorem which implies that all nerve equivalences are $D$-equivalences for any derivator $D$.
Mike,
clearly basic localizers have been introduced to localize Cat, no? ;-)
The entry should say a word about this. The natural question to address (at least to ask) is whether the weak equivalences of the canonical model structure and the Thomason model structure are basic localizers. Are they? (I’d need to think about the third axiom.)
@Urs - Thom model structure?
@Urs
It’s already stated on the page that the class of equivalences is not a basic localiser. I don’t really understand the intended applications of basic localisers either, but Grothendieck wrote (Pursuing stacks p.165):
These conditions are enough, I quickly checked this night, in order to validify all results developed so far on test categories, weak test categories, strict test categories, weak test functors and test functors (with values in $(Cat)$) (of notably the review in par. 44, page 79–88), provided in the case of test functors we restrict to the case of loc. cit. when each of the categories $i(a)$ has a final object. All this I believe is justification enough for the definition above.
The general context seems to be the problem of modelling homotopy types, so the intended examples are probably closely connected to the Thomason model structure. (What is nowadays presented as) the second axiom seems especially geometric in nature. I have not checked, but it seems plausible to me that, for each natural number $n$, the class of functors that induce $n$-connected morphisms of nerves should be a basic localiser.
Postscript. Maltsiniotis gives the same example on p.5 here, and also adds analogues for $n = -1$ and $n = -2$.
Thanks! That helps. (And sorry for dropping an ason, fixed now.)
I have used that information to give basic localizer an Idea-section and also added the Grothendieck quote after the definition.
One point as PS is not available in a finished, accepted version, beware of giving page numbers by themselves, rather give a section number and if you want to give a page number say what version (original, retyped,….) and what page numbering (in tiff version, original pages, pdf, …) the quote is from. There are a large number of tOC pages in the scanned version before AG’s page numbering starts.
In Cisinski’s “Le localisateur fondamental minimal”, Corollary 1.1.10 claims that any right or left adjoint is a weak equivalence (i.e. belongs to any basic localizer). But no proof is given, and I’m a bit stuck on proving it myself. He’s just observed that the projection $A\times I \to A$ is a weak equivalence, because the interval category $I$ has a terminal object, and by 2-out-of-3 it follows as usual for left homotopies that if a functor is related to a weak equivalence by a natural transformation then it is also a weak equivalence. But if $f\dashv g$, it seems all I can get from this is that $g f$ and $f g$ are weak equivalences, and I don’t see how to get from there to $f$ and $g$ being weak equivalences unless we have not just 2-out-of-3 but 2-out-of-6. Is there a trick I’m missing?
See Proposition 1.1.9 in Maltsiniotis’ book here. As you will see, the proof relies on a little more than the ingredients you describe; one needs the ’Quillen Theorem A’ type axiom.
Thanks! That’s actually a much simpler proof than going through the interval category. I wonder if that’s what Cisinski had in mind – if so, it seems odd to place it as a “Corollary” just after “Every functor homotopic to a weak equivalence is a weak equivalence.”
Yes, I agree, I’m not sure either just now; in ’Les préfaisceaux…’, Cisinki just cites Maltsinioitis’ book for this fact.
I’m not sure if you’re following this Denis-Charles, but long ago I wondered about the following. Any derivator gives rise to a basic localiser. Ordinary equivalences of categories induce a derivator in the obvious way (from the folk model structure, if you like; even though it is not cofibrantly generated, everything still works). Is the basic localiser coming from this derivator the minimal one? If so, this would give a to my mind rather beautiful way to construct homotopy theory from the starting point of category theory!
1 to 14 of 14