Timeline for The homotopy theory presented by a Waldhausen category
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18 events
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| Sep 11, 2019 at 20:52 | comment | added | Tim Campion | @BenWieland Ah, right -- I see now how confused I was! Thanks. | |
| Sep 11, 2019 at 19:58 | comment | added | Ben Wieland | No, the $K$-theory of finite complexes is the $K$-theory of finite spectra is $K(S)$ is big, rationally equivalent to $K(\mathbb Z)$. This shouldn't surprise you; this is exactly where you thought that the counterexample should be: not every map of finite sets is equivalent to a cofibration. | |
| Sep 11, 2019 at 16:42 | comment | added | Tim Campion | I see the confusion -- I didn't say how to compute $K$-theory from a finitely-cocomplete infinity category. Implicitly, I'm thinking one should take the cofibrations to be all maps and then just apply the $S_\bullet$ construction. | |
| Sep 11, 2019 at 16:40 | comment | added | Tim Campion | @BenWieland Actually, I think the $K$-theory is the same, by Barratt-Priddy-Quillen. That is, $K(FinSet,Mono,Iso) = K(FinTop,\text{all maps},Iso) = \mathbb S$ (where $FinTop$ is the $\infty$-category of finite cell complexes, so its iso's are the weak homotopy equivalences; $\mathbb S$ is the sphere spectrum). | |
| Sep 11, 2019 at 1:00 | comment | added | Ben Wieland | OK, but that just pushes around what the example of finite sets is a counterexample to. Your completion procedure expands it to finite cell complexes, which have a much larger $K$-theory. | |
| Sep 10, 2019 at 20:36 | comment | added | Andrea Gagna | With some saturation assumption, as @JoeBerner writes in his comment, the $\infty$-category you denote by $Wald(C, \text{cof}, W)$ looks pretty much alike to what Toen-Vezzosi denote by M(C) in A remark on $K$-theory and $S$-categories and to what Cisinski denotes by $\text{Ex}(C)$ in Invariance de la $K$-théorie par équivalences dérivées (see in particular the last section; the definition is given in 4.10). | |
| Sep 10, 2019 at 15:50 | comment | added | Joe Berner | I actually started thinking about this a bit a few days ago. You can look at Blumberg-Gepner-Tabuada for a relationship between Waldhausen's $S_\bullet$ construction of $K$-theory and the $S_\bullet$ construction for $\infty$-categories, BUT as far as I can tell they need some assumptions on the Waldhausen category (namely DKHS-saturation) that don't hold for this particular Waldhausen category. | |
| Sep 10, 2019 at 15:42 | comment | added | Joe Berner | I briefly commented previously and deleted it but Waldhausen's construction actually takes simple maps as weak equivalences. A simple map is a map where the preimage of any point has trivial Čech shape. Simple maps are simple homotopy equivalences and generate simple homotopy equivalences under zig-zags, but the definition of a simple homotopy equivalence actually gets messy and seems to be hard to work with. Simple maps do not satisfy 2/3, an example is $\Delta^0 \to \Delta^1 \to \Delta^0$. This is given in "SPACES OF PL MANIFOLDS AND CATEGORIES OF SIMPLE MAPS", look just after Prop'n 2.1.3 | |
| Sep 10, 2019 at 11:58 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Sep 10, 2019 at 3:24 | comment | added | Tim Campion | @BenWieland I'm confused. On p. 327 of Algebraic K-theory of Spaces, Waldhausen gives the simple homotopy equivalences specifically as an example of weak equivalences which don't satisfy his "saturation axiom" (which is just 2/3). I also don't see how what you're saying is a counterexample to (2) -- you give two examples of Waldhausen structures on $FinSet$ with different cofibrations, but in (2) I'm not asking whether $K$-theory is independent of $cof$ -- I'm asking whether it depends only on the $\infty$-category $Wald(C,cof,W)$, which depends on $cof$ as I've defined it. | |
| Sep 10, 2019 at 2:42 | comment | added | Ben Wieland | Simple homotopy equivalences do satisfy 2/3. But inverting them them inverts all homotopy equivalences see... A counterexample to 2 is that the category of finite sets is, technically, cocomplete as an $\infty$-category. If you take $S_\bullet$ for all morphisms you get zero, whereas $S_\bullet$ of monos is the sphere. But I'm not sure that this is really an interesting example; maybe one should just exclude it from the theory entirely. | |
| Sep 8, 2019 at 21:10 | comment | added | Tim Campion | Regarding Question 2, I'm now skeptical that the answer should be affirmative unless $(C,cof,W)$ has the property that every morphism is weakly equivalent to a cofibration, which is essentially the type of hypothesis used to show the cofibrations don't matter in the literature... After all, the $S_\bullet$ construction cares which morphisms are "cofibrations". | |
| Sep 8, 2019 at 21:08 | comment | added | Tim Campion | @BenWieland Good point. I'm confused about what lesson to draw. On the one hand, I might think that since simple homotopy equivalences don't satisfy 2/3, one shouldn't just localize at them. But then as you say, the space of simple homotopy equivalences is known to be interesting, so apparently it is interesting to localize at them... | |
| Sep 4, 2019 at 12:29 | comment | added | Andrea Gagna | I think the review of modern $K$-theory given by Robalo in his thesis §7.1 is very well-done and might answer (positively) to most of your questions. | |
| Sep 4, 2019 at 0:50 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Sep 4, 2019 at 0:47 | comment | added | Ben Wieland | Waldhausen categories are a ridiculously slick formalism solving a huge number of problems, like not having structured ring spectra. If you're trying to do $K$-theory, then, yeah, you should probably just work with ∞-categories. But Waldhausen was trying to relate $K$-theory to geometry. I think that there is a Waldhausen category where the equivalences are simple homotopy equivalences. The nerve of these equivalences is a space of interest. But it doesn't seem to be the maximal ∞-groupoid of an interesting (cocomplete) ∞-category. | |
| Sep 3, 2019 at 23:49 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Sep 3, 2019 at 23:43 | history | asked | Tim Campion | CC BY-SA 4.0 |