Comments K-theory notes Chapter I

Hi Ferdinand,

This is some longer list of typos/comments on your K-theory notes. (For now it's only chapter I; I might also go through the later chapters in this much detail in the semester break.)

Let me start with: I think it is really nice that you included proofs of I.42, I.47, I51 and I.52, since these are very important and useful results!

I was somehow a little confused by an apparent asymmetry in the proof of I.42. The dual proof uses right fibrations, and I was expecting the undualized proof to use left fibrations, but this doesn't work since the version of lemma 42a) for left fibrations still produces right adjoints p* and q with an equivalence f^p* ~ qg^, and not left adjoints as would be required for the undualized statement. But I guess it was simply a choice to use the right fibration version lemma 42a) and we could have used the left fibration version as well in your proof for the dual of I.42. And then for the undualized proof we would not use the second Yoneda embedding of D^op into Fun(D,An) = P(D^op) but rather the embedding of D^op into Fun(D,An^op) ~ P(D)^op, right? I guess the fact that An is involved makes me a little confused about how to dualize. Do you have any comments about this?

Now a long list of comments/typos/questions per page:

p.9, I.9 The pullback diagram defining Hom_C(a,b) is in fact also a pullback in the infty-category Cat_infty, right? We didn't consider this relevant at this point in the lecture, but if I'm not mistaken we do use this several times later in the course without ever justifying, so perhaps you can say a word about this at this early stage. Does it suffice to observe that (s,t): Fun(Delta^1,C) -> C x C is an isofibration between qcats and thus a fibration in the Joyal model structure?
p.11, I.15(e) For emphasis it could perhaps be mentioned that this second K(R) does have to do with K-theory.
p.12 the "tu" should be "to".
p.13, I.21: I guess also here it is useful to already remark that this pullback diagram can be seen as a pullback in Cat_{\infty} since fgt is a left fibration and thus an isofibration.
p.13, I.23 Maybe for completeness you can clarify the later notations 'St^{cocart}' and 'St^{cart}'.
p.14, I.24(a) Your description suggests that one cannot really define homotopy pullbacks without infty-categories. Maybe you could add that it is also possible to make this precise in other ways, for example in the way that Fabian did last semester.
p.14, I.24(b) The "Now replacing Hom_E(x',y')" should be "Hom_E(y,y')".
p.14, In "if it is a cocartesian fibrations", it should be "fibration".
p.15, I.24a. The "it is no isofibration" should be "it is not an isofibration".
p.16, I.25b. Here you have proved something even stronger: t: Ar(C) -> C is a cartesian fibration if and only if C has pullbacks.
p.18, I.26(b): Unfinished sentence: "construction of ...".
Also it should be TwAr(C) -> C^op x C.
p.21, I.33(b) The "F const" should be "const".
p.22, below I.35a: Here "no inner fibration" should be "not an inner fibration". Also, why don't we even need to require it to be an isofibration?
p.24, I.38 Does the following argumentation also work? The diagram already consists of cofibrations so the pushout in sSet (the 2-spine I2) is also a homotopy pushout, so by I.34 we obtain that the colimit in Cat\infty is a fibrant replacement of I_2, which is Delta^2 since the inclusion of I_2 into Delta^2 is inner anodyne and thus a categorical equivalence.
p.26, I.42 The "and same for D" should be "and same for E".
p.28, I.44 More accurate would perhaps be to say 'preserves and detects the existence of a limit'.
p.29, I.47 I don't quite get why we need to talk about model structures here. By I.32, we just need to check that Fabian's description does indeed give a left adjoint object, which seems to be the step that you hid away under " a quick unraveling shows that Lf! can indeed be described as above." The fact that you don't spell this out also suggests to me that you have a quicker way to see this than what I did. I thought: we want to show that a left adjoint to the inclusion Right(D) -> Cat/D is given by factoring a functor E -> D as a cofinal map E -> E' and a right fibration E' -> D. So if p: E' -> D and q: F -> D are right fibrations and i: E -> E' is a cofinal map, we need to check that Hom{Cat/D}(p,q) ~ Hom{Cat/D}(p \circ i, q). For this, we need several results from the previous semester about hom-anima in overcategories and we need XI.28 that says that the cofinal map E -> E' is up to equivalence just a right anodyne map so has the left lifting property (even in the \infty-category Cat) with respect to left fibrations.
p.31, I.50 You formulated the statement in terms of a functor I -> C rather than C ->D, but forgot to change this also in the proof.
p.31, I.50, bottom: isn't c/C -> C a left fibration?
p.32, I.50a, the "e in E" should be "e in E x_C {c}".
p.33, I.53, perhaps it would be good to include the functor f in the terminology, as in 'right Kan extension of F along f'.
p.33, Proof of I.52, It should say "by construction, Lan_f F is a left adjoint object". Also, can you clarify what you mean by "by construction" here? I suppose you mean "by the construction we did in the proof of I.42", because I don't see why it would follow from the the properties of the functor given in the statement of I.42.
p.35 The "object wise" should be "objectwise".
p.35, bottom: I guess here the left P(C) in the subscript should be Fun(C^op,Cat)?
p.35, bottom: and that the right Cart(C) should be Right(C)? (Well technically this isn't wrong but I guess changing it could save confusion.)
p.36, proof of I.51: The "G o Y^o" should be "G o Y^C" I guess.
p.36, I.55 Shouldn't we check that they agree as functors P(C)^op -> An. (Which is actually what you even do, but what you don't say we should.)
p.38, bottom: can you add an explanation of why cosk_n(X) works as a left adjoint object? I would honestly be surprised if this really gives a left adjoint object. What I've seen before is that one does an iterative construction where one keeps killing higher and higher homotopy groups by adding single cells for each generator. (And for this we can see that it gives a left adjoint since for n-truncated anima Y, every map S^i -> Y is trivial so every map X -> Y has (up to homotopy a unique) extensions over the attached i-cell.)
p.38, bottom: For the statement that the fiber is a K(pin+1(X),n+1), I think I might be missing something since I don't consider it obvious how "the above discussion" already contains a proof of this fact. What I have in mind right now are the following steps: (a) by a long exact sequence argument, the fiber only has non-zero homotopy group in degree n+1, given by pi(n+1)(X). Now we need to prove that if X has only pii in degree n, then it is an Eilenberg MacLane anima. (b) For such X, we have C(X) ~ pin(X)[n] by exercise 56a and Hurewicz (c) So we get a unit map X -> K(C(X)) = K(\pi_n(X)[n]) = K(pi_n(X),n), which is a weak homotopy equivalence and thus an equivalence in An. Did you also have this in mind, or did you see something shorter?
p.38, bottom: am I right that X is the limit of this tower in An? I guess we can argue: pi_n commutes with limits, and the sequence of pi_n's of all the truncations of X is eventually constant and equal to pi_n(X), so the canonical map from X to the limit is a weak homotopy equivalence.
p.38, bottom: unfinished sentence: it stratifies */An into...
p.39, bottom: I suppose here all the pullbacks are homotopy pullbacks, right? If so, you could perhaps clarify somewhere in your notes that you stop writing explicitly the \times^R as you did in the beginning.
p.40, I.57(a), the codomain of the adjunction should be D instead of C.
p.40, I.57(a) in the warning: the cosimplicial anima const_* should have domain category Delta instead of Delta^op.
p.41, I.58 Fabian's proof of part (a) is overcomplicated. One can simply consider the triangle with Hom_D(d,x) -> Hom_D(LRd,x) and Hom_D(d,x) --R--> Hom_C(Rd,Rx) ~ Hom_D(LRd,x) and show that it commutes by remarking that by Yoneda it suffices to check this for x = d on id_d, but on id_d we get the counit LRd -> d in both cases.
Also, I don't get why the order of (a) and (b) is reversed in the proof.
p.42, I.60(b) In the "in fact", the domain of p^op should be C^op rather than C.
p.43, Proof of I.59: the second reference to Lemma I.60(b) should be I.60(a).
Also, it should be Hom_{C[W^-1]}(-,px) rather than Hom_C(-,px).
p.43, end of proof of I.59: the "and R-local object" should be "an R-local object".
p.43, I.61 I guess it's just a matter of taste, but in Fabian's formulation of the proof he showed that L(colim RF) satisfies the condition for colim F rather than assuming colim F exists on the outset and showing it is equivalent to L(colim RF).
p.44, I.61a, bottom: are you sure the counit is once a left inverse and once a right inverse? To me it seems it is a left inverse in both cases. (But this suffices since we already know it is an equivalence.) Also, what I get is that it isn't the counit at Lx but rather L applied to the counit at x. Could this be the case?
p.45, I.62(b): were the quotation marks around ' "CW" approximation' meant to be put this way, or was it meant to be "CW approximation"?
p.45, I.62(c): is it worthwile pointing out that this is often called the 'derived \infty-category' to avoid confusion with the derived 1-category?
p.47, Proof I.63: To show that S \otimes^L_R R ~ S, you argue via Tor-groups, but I suppose we know these Tor-groups only because we can say 'R is already projective over R, so we can take R itself as projective resolution', and this argument also directly gives that S \otimes_R^L R ~ S, right?
p.47, Proof I.63: the square is a pushout square, not a pullback square.
p.47, I.64: X is actually a simplicial anima, not a cosimplicial anima.
p.48, I.64(c): the first arrow in the composition map should be (d0,d2). Also, in the very last line of (c) it should read "g in X_1^x" instead of "x in X_1^x".
p.48, I.65: I had a thought about the explanation of conditions (b) and (d) and I think we can say a little bit more about why they are morally saying the same thing. In the words of Emily Riehl: completeness says that all things that are categorically equivalent are identified, i.e. all categorical equivalences are identity morphisms. For condition (a) and (c) it is intuitive that this is what it is saying, and condition (d) is basically just the statement that all higher simplices having equivalences as edges are degenerate. To make it more intuitive why also condition (b) is saying the same thing, notice that this is just the higher categorical way of phrasing the requirement that identity morphisms are closed under the 2-out-of-6-property. This is also equivalent to all equivalences being identities since the equivalences in a category are the smallest collection of arrows containing the identities and closed under the 2-out-of-6-property (compare also Lurie, Proposition 5.4.6.6 or Tag 01XV of Kerodon.)
p.49, proof of I.64: I tried to prove that the completeness condition (a) is equivalent to the rest, but somehow failed. Proving (a) <-> (c) seems hard since the data in (c) doesn't provide coherent inverses. For proving (a) <-> (b), it would perhaps suffice to see that J is the pushout in Segal spaces of \sqcup <-- Delta^1 \sqcup Delta^1 --> Delta^3, but I am not sure whether this is even true.
p.49, proof of I.64: in the big diagram at the bottom, I think the argument for why this square is a pullback in An is that the original square is a pullback in Cat_\infty (see the remark to I.9 above) and that core is a right adjoint and thus commutes with limits. Perhaps what you are arguing also works, but it confuses me a little since it looks like a pointset-level argument and the pullback we want to take should be a homotopy pullback and not a pointset-level pullback of simplicial sets.
p.49, proof of I.64: right now you have not yet said anything about why the essential image of the Rezk nerve functor should be all complete Segal spaces. Fabian said in the lecture: "one can argue similarly for the unit X --> N^r(asscat(X)) when X is a complete Segal anima", and this does indeed show that every complete Segal anima is in the essential image of N^r.
p.49, I.66: perhaps it is useful to emphasize that we take the homotopy colimit of T after regarding it as a functor from Delta^op to sSet rather than to Set.
p.50, I.66: I'm a little confused by your use of [n] in your application of I.51. You specifically seem to say that you want to reserve [n] for the quasi-category and Delta^n for the (discrete) simplicial anima. So then in this case we should argue that both sides send Delta^n to , rather than [n] to , right? Or am I misunderstanding something?
p.50, I.67: do you mean "full" subcategory rather than "complete" subcategory?
p.50, I.68(a): you omitted a remark that Fabian made in the lecture, namely: "But we have asscat(N(C)) ~ C from the description of asscat for Segal anima, and thus N^r(C) is the completion of N(C)." The fact that you omitted it makes me think it might be incorrect, so let me just write my thoughts and perhaps you can comment on it. It is clear by I65.2 that the composite asscat(N(C)) -> asscat(N^r(C)) -> C is an equivalence on hom-anima, so it would remain to see whether it is essentially surjective. By I65.1 it suffices to show that |N(C)^x| -> |N^r(C)^x| ~ core(C) is an equivalence. But N(C)^x is given by [n] \mapsto {(f_1, ..., f_n) | composable sequence of n isomorphisms in C}, which is the ordinary nerve of the 1-category core(C). But by remark I.66, the realization of N(core(C)) seen as a functor Delta^op -> Set -> An is N(core(C)) itself, seen as an anima, since the simplicial set N(core(C)) = core(N(C)) is already a Kan complex. So if I'm not mistaken this gives that |N(C)^x| ~ core(N(C)) as anima, right?
p.50, I.68(c): I'd recommend avoiding the notation "const. sAn", since I think it is confusing with the space in the middle. An alternative would be sAn^const.
p.51, I.69, bottom: I think that instead of {1, ..., n} it should be {1, ..., n+1}, right? Otherwise it would give [n] rather than [n+1]. In Fabian's lecture, he did indeed go to n but that was because he used *, 0, 1, ..., n rather than 0, 1, 2, ..., n + 1.
Also, why do we put An and /An in the diagram and not core(An) and core(/An)?
p.51, I.69, bottom: you write that we need that core commutes with pullbacks in step (3), but the core(-) is already outside of the pullback to begin with after step (2) so strictly speaking we needed this argument in step (2).
p.52, I.69: it seems like you're saying that the inclusion Cat1 -> Cat\infty is fully faithful, but it is not: the hom-anima in Cat_\infty between 1-categories may contain non-trivial natural equivalences and are therefore different from the discrete anima of functors. But if I'm honest, I think that even if I would assume that it is given by this discrete anima, I wouldn't know how to finish the argument. Actually, I'm not even sure how we defined the join of simplicial anima.
p.52, I.70: the codomain of Q(C) should be Cat_\infty rather than C. Furthermore, it is a simplicial infty-category rather than a cosimplicial one (2x).
p.52, I.71: the word "spams" should be "spans".

Finally here is a list of (very pedantic...) general comments about some of the language:

you often write 'a infty-category' instead of 'an infty-category';
you're not consistent in your use of 'subcategory' versus 'sub-infty-category'. (Personally I like 'subcategory' better'.)
I'm not really the best person to judge this but I think you're mixing up British spelling with American spelling when you're using 'fibre' instead of 'fiber' on the one hand, but also '-ization' instead of '-isation'.
you often write "left-adjoint" instead of "left adjoint" (I guess this is on purpose, but I’ll mention it anyway since it seems non-standard.)

Hi Bastiaan,

I can't thank you enough for your enormous list of suggestions. It really improved the quality of these lecture notes. In the current version of the script I credit you with the proofs you suggested (by small comments like "Bastiaan has pointed out a simpler proof that doesn't need model categories") and I'm also planning to add an "Acknowledgments" point to the introduction to thank you, Fabian and solov-t (whose actual name I'll hopefully manage to find out) four your help. Is it ok for you (and I probably should have asked that earlier) to have your name mentioned in the script?

Regarding your comments/suggestions/questions:

The proof of I.42 confused me too. Initially, I had expected that the undualized statement should be proved by applying the other Yoneda embedding and proving a version of I.42a for left fibrations and the left adjoints p! and q!. It turns out, however, that the the undualized proof is way more counterintuitive (which is the whole reason I prove the dual statement). You are right that we would need to use the "opped Yoneda" embedding D^op --> P(D)^op, since neither of the Yoneda embeddings preserves colimits. Your comment also made me realize that the formulation "the undualized version follows by applying (-)^op everywhere" is potentially misleading: I wouldn't recommend trying to undualize the proof of I.42; instead, one should prove the undualized statement by applying the dual statement to p^op: E^op --> C^op. I tried to make that point clearer in the latest version of the script. You're also right that the version of I.42a for left fibrations is still about right adjoints p* and q*, and that I could have used the left fibration version as well. When I TeXed the proof of I.42, I decided against the left fibration version because I wanted to be able to refer to the proof of I.47 without having to switch between the covariant and the contravariant model structures (which is now obsolete, see below).

Incorporating your suggestions changed the page numbering, so I'll include both the old and the new page numbers (where always old ≤ new):

p.9/p.9, I.9: You're right, the pullback is also one in Cat_infty, the reason being that (s,t) is an isofibration and I think we also need that everything is fibrant in the Joyal model structure (see [Higher Categories II, Corollary VIII.52]). I agree this should be mentioned, but after I.9 seems a bit early, so I made a new Example* I.35a on p. 22, right after Theorem I.34 which ensures that limits in Cat_infty can be computed by homotopy limits in sSet.
p.11/p.11, I.15(e): I agree, added a comment in the latest version.
p.12/p.12: Fixed.
p.13/p.13, I.21: I also put this into Example* I.35a.
p.13/p.13, I.23: I added a small comment about the cartesian version and introduced the notations "St^cart/cocart" and "Un^cart/cocart".
p.14/p.14, I.24(a): You're right, the way I wrote this gives the wrong impression. I added a reference to Fabian's definition of homotopy limits last semester and to Theorem I.34, which connects them to limits in infinity-Categories.
p.15/p.16, I.24b: Changed "Hom_E(x',y')" to "Hom_E(y,y')".
p.14/p.14, I.24(c): Changed "fibrations" to "fibration".
p.15/p.15, I.24a. Changed "it is no isofibration" to "it is not an isofibration".
p.17/p.17, I.25b: If I remember right, Fabian formulated the exercise in the weaker way, because he wasn't quite sure whether the converse holds as well. But your comment convinced me that giving the stronger formulation and thus not being faithful to the lecture might prevent some confusion, so I changed that.
p.18/p.18, I.26(b): Finished the unfinished sentence and added a "^op x C".
p.21/p.22, I.33(b): Changed "F const" to "const".
p.22/p.23, below I.35a: I think you're right that we need it to be an isofibration.
p.24/p.25, I.38: Again, I suspect we also need that all objects are cofibrant, but that argument should also work. If I remember correctly, Fabian even commented that his way of computing the pushout is not very efficient and he choose it just to show that I.36 can sometimes be computable. Anyway, I've added your method to compute the pushout.
p.26/p.27, I.42: Changed "and same for D" into "and same for E".
p.28/p.29, I.44: Changed the formulation to "precomposition with alpha preserves limits and detects their existence".
p.29/p.31, I.47: I didn't realize that we can actually prove I.47 with the methods we have available (plus some facts about cofinal maps). Thanks for pointing that out! I decided to include your proof instead instead of my original one, which I cut down to three sentences, moving the model category arguments to I.42a instead. I'm not sure if one can prove I.42a too without referring to the contravariant model structure, and to be honest I'm too afraid to try because I don't want to get involved with any concrete description of the right adjoint p_* :stuck_out_tongue: . I know Lurie gives a simple description in [HTT, before Prop. 4.1.2.8], but I'm not sure if it really is correct.
If I'm not mistaken, it should really be easy to see that Lf! is described as in I.47: We know that f! simply sends p: E --> C to E --> C --> D. Now Lf!(p) is given by f!(cofibrant replacement of p). But everything is cofibrant in the contravariant model structure, so this is just f_!(p) = E --> D. Finally, the equivalence (sSet/D)_infty ~ N^c((sSet/D)^cf) sends E --> D to a fibrant replacement. Since every cofinal map is the composition of a trivial fibration and a right anodyne map, any factorization E --> E' --> D into a cofinal map followed by a right fibration will do.
p.31/p.33, I.50: Fixed.
p.31/p.33, I.50, bottom: Yes, c/C --> C (now i/I --> I) is a left fibration. I fixed that.
p.32/p.33, I.50a: Changed "e'' in E" to "e'' in E x_C {c}".
p.33/p.34, I.53: I agree, f is now included in the terminology.
p.33/p.35, Proof of I.52: You're right, it's not at all clear what I mean by "by construction", and the fact that I prove the dual statement in I.42 makes it even worse. In the updated version I give a thorough argument.
p.35/p.37, proof of I.53b: Changed "object wise" to "objectwise", the left P(C) to Fun(C^op,Cat_infty), and the right Cart(C) to Right(C).
p.36/p.38, proof of I.51: Fixed.
p.36/p.38, proof of I.55: Changed "C^op --> An" to "P(C)^op --> An".
p.38/p.40: It should be cosk_{n+1} instead of cosk_n, but otherwise this should work. Intuitively, taking the coskeleton should be just a really brutal way of killing all homotopy groups, by simply filling all possible holes. See Charles Rezk's answer to this MO question (and I think Fabian also mentioned this in the lecture). But it's worth spelling out that iteratively killing higher and higher homotopy groups is really all that happens, so I added that in the new version of the script.
Your argument that the homotopy fiber is a K(pi_n+1(X), n+1) can be somewhat simplified using the facts we already established in the discussion after the Eilenberg-MacLane theorem. I added some more details.
Yes, a connected CW complex is weakly equivalent to the limit of its Postnikov tower, see [Hatcher, Corollary 4.68] for example. I thought about whether there is a quick proof with our methods, but there seems to be a problem: I'm a bit skeptical whether pi_i=pi0 Hom/An((S^i,),-) preserves limits. Hom_/An((S^i,),-) certainly does, but pi_0 is only a left adjoint. I think it should be possible to show that it commutes with this particular limit, but I doubt that this works in general.
I finished the unfinished sentence.
p.39/p.41 bottom: Yes, these pullbacks are taken inside An. I think the most sensible point to drop the ^R would be after Corollary I.50, since this is the last ingredient in setting up the infinity-categorical machinery of limits and colimits. So I added a corresponding announcement there.
p.40/p.42, I.57(a): Fixed both typos.
p.41/p.43, I.58: Very nice argument! I replaced Fabians proof with yours, and gave you the credits you deserve.
I also don't know why Fabian swapped (a) and (b) in the lecture, but I decided to keep it that way.
p.42/p.44, I.60(b): Fixed.
p.43/p.45, Proof of I.59: Fixed the reference and the two typos.
p.43, end of proof of I.59: the "and R-local object" should be "an R-local object".
p.43/p.45, I.61: I've reformulated the last sentence in the colimit part to be more mathematically precise. Also I realized that the last two steps in the limit calculation don't make any sense and fixed them.
p.44/p.46, proof of I.61a, bottom: I think you are correct that it is a left inverse in both cases, but I think the counit applied to Lx should be correct. The first triangle identity says that L =(L unit)=> LRL =(counit L)=> L is the identity on L, so the counit applied to Lx should be the correct left inverse of (L unit). Unravelling the second triangle identity should yield the same for (unit L), although it is a bit of mindfuckery since R is surpressed in the notation of I.61a.
p.45/p.47, I.62(b): The quotation marks were put at the wrong position and had no deeper meaning :wink: .
p.45/p.47, I.62(c): In the new version D(R) is called the derived infty-category, and I added a remark that its homotopy category is the usual derived 1-category.
p.47/p.48, Proof I.63: You're right, the Tor computation is unnecessary/circular, and so I ditched it.
p.47/p.49, Proof I.63: Just to be a smartass: It is a pullback square as well since D(R) is stable :nerd_face: ... but of course "pushout" is what I intended to write.
p.47/p.49, I.64: Changed "cosimplicial" into "simplicial".
p.48, I.64(c): Fixed the two typos.
p.48/p.50, I.65: Thank you (and Emily Riehl) for this wonderful explanation! This is so much better than what I wrote before. I scrapped my own explanation and basically copied your words into the script, with only Fabians picture for (b) added (which fits smoothly into the rest).
p.49/p.51, proof I.64: I didn't attempt to prove the equivalence of (a) to (d). I might do so when I find the time in the semester break, but no guarantees.
p.49/p.52, proof of I.64: I think we used the fact that a pullback of infty-categories with one leg an inner fibration induces a pullback on cores at some point in the beginning of Higher Categories II. At least it should be true if one leg is an isofibration. But I agree, the cleaner argument here is that the Hom-Pullback can also be taken in Cat_infty, so I changed that and added a reference to the nex Example* I.35a.
I must have forgot to write down Fabians argument for the essential image of N^r.
p.49/p.52, I.66: I clarified what the "homotopy colimit of T" is supposed to mean.
I wrote [n] instead of Delta^n since I thought of it as the element of Delta before the Yoneda embedding Delta --> sAn. But you're right, I should rather write Delta^n to avoid confusion. Also the remark about when to use [n] and when to use Delta^n seems a bit misplaced, so I moved it to the beginning of the section.
p.50/p.52, I.67: The complete Segal anima everywhere must have confused me, I meant to write "full subcategory".
p.50/p.53, I.68(a): Apparently I forgot to include Fabian's remark, so I fixed that and added your explanation (which I think is correct).
p.50/p.53, I.68(c): I struggled to come up with a good name (I thought about CsAn, but that was out of the question as soon as CSAn was introduced), so I went with a shitty name. But I like your suggestion, only I changed it to sAn_const.
p.51/p.54, proof of I.69: You're right, it should be {1, ..., n+1} and core(*/An), core(An). I rewrote the proof of I.69, adding another step to make it more understandable.
p.52/p.55, proof of I.69: I've completely overlooked this subtlety, so thanks for pointing it out! In this case, however, it should be fine that Hom_Cat_infty([m],[n]^op join [n]) is discrete: Using that Fun(N(C),N(D)) = N(Fun(C,D)) for 1-categories C, D, we see that Hom_Cat_infty([m],[n]^op join [n]) is the core of the 1-functor category. But every natural isomorphism of functors to [n]^op join [n] must be an identity because [n]^op join [n] has only identities as isoorphisms (by inspection). I also don't know how to define joins of simplicial anima in general, but we can just form (Delta^n)^op join Delta^n as a join in sSet and consider it as a discrete simplicial anima, and then the arguments just work. In the new version of the script, I added some clarifying remarks.
p.52/p.55, I.70: Fixed. This damn "op" in "Delta^op" really confuses me every time.
p.42/p.55, I.71: Fixed.

Also thanks for your language style remarks! I'll definitely keep them in mind in the future, but I think it's too late to try to achieve consistency in the script so far ...

I'm looking forward to your extensive list of comments for Chapter II! (No pressure of course :stuck_out_tongue: .)

Hi Ferdinand,

Yes, I'm fine with having my name mentioned in the script. Also, I'm pretty sure that solov-t is Thiago in case you want to ask him next Thursday.

p.31, I.47: First, about the sentence "a quick unraveling shows that Lf! can indeed be described as above": right, of course, that makes a lot of sense. I kinda skipped over the model theory part and I incorrectly thought that the sentence was meant to say "now that we know there exists a left adjoint object, a quick unraveling shows that this left adjoint object can indeed be taken as above" rather than "the above does indeed describe the left derived functor of f!".
Second, I'd like to make another suggestion about the proof of I.47. Your current proof essentially proves two left adjoints at the same time: you prove that 1) there is an adjunction f!: Cat/C <-> Cat/D: f^ given by composition with f and pullback, and 2) the inclusion Right(D) -> Cat/D has a left adjoint given by factoring a functor E -> D as a cofinal map followed by a right fibration. I somehow regard these as two separate arguments, and also think that it might be useful to see the formation of the left adjoint object really as two steps: 1) we first compose p with f to get a category over D, as postcomposing with f is left adjoint to f^ on the level of overcategories, 2) and we then need to 'universally' replace this category over D with a right fibration over D. I guess one argument in favor of this distinction is that step 1) is really not so much a statement about the composition p circ f being a left adjoint object under the pullback functor f^*, but rather it is a statement about the very meaning of f^*, namely that it is a right adjoint to the composition functor Cat/C -> Cat/D.

Now that I think about it, I'm not sure whether we ever properly defined what f^*: Right(D) -> Right(C) is, and perhaps this deserves a discussion on its own which could already include step 1). I'll simply write it out (but since I'm realizing this is quite long, feel free to choose yourself whether you consider it worth reorganizing I.47.) There are three ways I can now think of to interpret f^*: a) it sends a right fibration p' to the pullback f^*(p') in Cat\infty, b) it sends a right fibration p' to the pullback f^*(p') in sSet, c) it corresponds under Right(C) ~ Fun(C^op,An) to precomposing with f^op: C^op -> D^op. Since we are pulling back a right fibration, it is clear that a pullback in sSet is also a pullback in Cat\infty, so b) is in particular an instance of a). For description c) it is immediately clear why it forms a functor, and by the description of the unstraightening Un: Fun(C^op,An) -> Right(C) in terms of pullbacks in Cat_\infty (remark 35a) it is also immediately clear that c) corresponds to a) under the unstraightening equivalence, so that a) also forms a functor.

We can now consider the composition functor f!: Cat/C -> Cat/D, and consider the functor f^*: Cat/D -> Cat/C that sends p: E -> D to the pullback f^*(p): f^*(E) -> C. This forms a functor by I.32, since the pullback f^*(p) is essentially by definition a right adjoint object to p under f!: Cat/C -> Cat/D. Also, by description a), it restricts on right fibrations to the functor f^*: Right(D) -> Right(C).

Everything up to this point was probably already assumed to be clear when Fabian wrote "the pullback functor f^*: Right(D) -> Right(C)" in the Theorem, and if not, I think it deserves to be treated before discussing the question of its left adjoint. If we assume this discussion, then the construction of the left adjoint becomes clearer in my opinion by separating it in the steps mentioned above. So explicitly, I'm thinking of:

By I.32 we can construct a left adjoint objectwise.
Since Right(C) -> Cat/C is fully faithful, it suffices take a right fibration p: E -> C and find a left adjoint object under the composite f^*: Right(D) -> Right(C) -> Cat/C, which as we have discussed is the same as the composite Right(D) ---incl--> Cat/D ---f^*---> Cat/C.
Since 'left adjoints compose', we can form this left adjoint object in two steps. For f^*: Cat/D -> Cat/C we have a left adjoint given by precomposition with f, again just by how we defined f^*. Therefore, it remains to understand how to construct a left adjoint to Right(D) ---> Cat/D.
Now finally we claim that this can be done by factoring an arbitrary functor E -> D as a cofinal map E -> E' followed by a right fibration E' -> D. Here we now finally do need some pointset-level discussion of functor categories as you wrote down in your proof, but my hope is that it now becomes more readable since there are no annoying appearances of f^* that obscure what is really happening here.

Having discussed this, I'd like to also discuss the issue of n-truncation and the n-th coskeleton a little more. Since Charles Rezk says it, I guess it should be true (argument by authority) but to be honest I still haven't wrapped my head around why it is true. Something that is easy to read off is that cosk_n(X) is n-truncated, but to me it is quite surprising that this is the universal (n-1)-truncation since the defining universal property of cosk_n(X) is just so unrelated to this. Something that also confuses me a lot right now is why cosk_n(X) is (n-1)-truncated and not just n-truncated. It seems to me that for the homotopy groups in degree k <= n, we need to consider maps into X out of the objects sk_n(\partial Delta^k) -> sk_n(Delta^k), but both sides are already n-skeletal and thus this map is just the same as the map \partial Delta^k -> Delta^k, indicating that pi_k(cosk_n(X)) = pi_k(X) for k <= n. What am I doing wrong?

But anyway, even if I would have done these calculations right and I obtain that cosk_n(X) is (n-1)-truncated and the map X -> coskn(X) induces isomorphisms on the homotopy groups for k <= n-1, this wouldn't prove the existence of a left-adjoint object tau{<=n-1} right? We need to prove via other methods that this left-adjoint object tau{<=n-1}(X) exists (for example by the argument that literally kills the homotopy groups iteratively), and that its homotopy groups are also given by the homotopy groups of X in degrees <= n-1 and 0 for all degrees >= n, and only then we can conclude that the canonical map tau{<=n-1}(X) -> cosk_n(X) is a weak homotopy equivalence and thus an equivalence in An.

Or is it actually possible to show that cosk_n(X) satisfies the desired universal property directly? If so, I would be highly appreciative of a reference, since online I cannot find this claim being proven anywhere...

Finally some remaining comments:

p.40, there is now an unfinished sentence "...side induces an isomorphism on the left-hand side, as shown in."
p.46, I.61a) You were right, I had myself confused there.
p.48/p.50, I.65: Now that you've really copied the words I claimed Emily Riehl has said and put quotation marks around it, I felt obliged to go back to the video and find out the exact words she said (what I wrote was essentially just me paraphrasing her from memory, and my memory is not good enough to reproduce exact quotes haha). Her exact words are "the completeness condition says that isomorphisms are equivalent to identities". Since I don't want to put my own words into her mouth, maybe you could replace the current quoted sentence by this new one, which is perhaps nicer anyway. I should perhaps also clarify that, even though her explanation helped me a lot in understanding complete Segal spaces, the context in which she said it wasn't directly about complete Segal spaces, but rather about a way of formulating \infty-categories in Homotopy Type Theory (indirectly it was about complete Segal spaces though, since the underlying model in ZFC used for proving consistency of her version of HoTT is in fact given by complete Segal bisimplicial sets). For a reference to where she said these words, see https://youtu.be/A6hXn6QCu0k?t=5294.
p.52. I.68(a): good to see that the true words of Emily Riehl fit here even better than my version of them :D

Hi Bastiaan,

thanks again for your suggestion regarding I.47! I ended up splitting the proof into several steps as you advised, and I hope this makes it less messy. The proof that Cat\infty/C --> Cat\infty/D is left adjoint to f^*: Cat\infty/D --> Cat\infty/C is still a bit cumbersome though, which is a bit of a shame given that this would become trivial using (the dual of) Corollary I.50, but we can't use that without running into a circular argument. (You probably won't see any changes in the script yet, see below.)

Regarding our old friend, the coskeleta: I think what confuses you is that Hom_sSet((Δ^n, ∂Δ^n), (X,x)) is indeed the same as Hom_sSet((Δ^n, ∂Δ^n), (cosk_n X,x)), but once we pass to homotopy classes, all elements of the latter will be killed! Let me expand on this a bit and sketch an argument why cosk_n X really has the right homotopy groups. I'll use the equivalent description of pi_k-1 as homotopy classes of maps (∂Δ^k, 0) --> (X,x). Using Kan complex magic, such a map is 0 in pi_k-1 iff it extend to a map (Δ^k,0) --> (X,x). For k <= n+1 we see that maps (not homotopy classes) (∂Δ^k, 0) --> (X,x) are in bijection with maps (∂Δ^k, 0) --> (cosk_n X,x), hence pi_k-1(X,x) --> pi_k-1(cosk_n X,x) is surjective for k <= n+1. However, it is only injective for k <= n since only then also maps (Δ^k,0) --> (X,x) are in bijection with maps (Δ^k,0) --> (cosk_n X,x). For k >= n+1, we see instead that sk_n(∂Δ^k) = sk_n(Δ^k), hence maps (∂Δ^k, 0) --> (cosk_n X,x) are the same as maps (Δ^k, 0) --> (cosk_n X,x), which means that pi_k-1(cosk_n X,x) vanishes for k >= n+1, which is what we want (after removing the index shift in pi_k-1). However, you're completely right that cosk_n alone won't help us to construct the (n-1)-truncation functor, or at least I don't see any direct argument for that. What one should do instead is basically what you described: If Y is already (n-1)-truncated, then the argument above shows that Y --> cosk_n Y is a homotopy equivalence. Hence, to show Hom_An(cosk_n X, Y) ~ Hom_An(X,Y), we may replace Y by cosk_n Y. But then I believe Fun(X, cosk_n Y) = Fun(cosk_n X, cosk_n Y) holds even as an isomorphism of simplicial sets. I've clarified (although it isn't online yet) that the truncation functor is really given by attaching cells to kill homotopy groups, and that it just so happens that cosk_n is a way to do so.

Also thanks for the correct quote and the link to Emily Riehl's video! When I copied your explanations, I wasn't quite sure whether to put your recollection of her quote into quotation marks or not, but then I thought: "Well, why not, no one will probably care." But of course it's better to have the precise quote, and now the script (which isn't online yet) also has a clickable link to Emily Riehl's video!

It'll probably take some days before the script is updated. I'm currently still working on organizing Fabian's crash course on E_\infty-ring spectra, and also I'm thinking about what might be another flaw in the construction of Day convolution. If you are interested (perhaps you can resolve my concerns): For the Segal condition to hold, it seems to me we would like that Fun(C, O_1)^n --> Fun(C^n, O_1^n) (which is an injection of simplicial sets, but not necessarily fully faithful) is an equivalence onto its essential image. However, this isn't true for arbitrary injections of quasicategories. Functors with this property should satisfy a certain pullback condition (see the nLab articles about replete subcategories and pseudomonic functors -- nLab does it for 1-categories only, but it should work in infinity-Land as well). And I don't see why this pullback condition should be satisfied in our case.

Have a nice weekend!

About the coskeleta: of course, I completely forgot taking homotopy classes. Thanks for your explanation!

EDIT: I think that what I say below is nonsense. I am mixing up the categories C^\otimes and its underlying category C_1 = C... I'll leave the original comment here in case it's still helpful:

_For your concern about Day convolution: I think it is actually true that for categories C and D with C symmetric monoidal, the "take products" functor Fun(C,D)^n -> Fun(C^n,D^n) is fully faithful. We have an equivalence Fun(C^n,D^n) ~ Fun(C^n,D)^n by sending F to the collection of functors pr_i o F and under this equivalence the above functor sends the tuple (F1,...,Fn) first to F1 x ... x Fn and then to the tuple of functors pr_i o (F1 x ... x Fn) = F_i o pr_i: C^n ---pr_i---> C ---F_i---> D. Since hom-anima in functor categories are computed componentwise, it remains to show that for every i the functor Fun(C,D) -> Fun(C^n,D): F \mapsto F \circ pi_i is fully faithful. For this, in turn, it suffices to show that the functor const: D' -> Fun(C,D') is fully faithful for any \infty-category D' (apply this to D' = Fun(C^i,D) for all i < n.) We proved last semester that this is the case if and only if |C| ~ . But isn't it true that the unique object in C_0 ~ is a final object in C as a consequence of the third operad condition? (As <0> is the final object in Gamma^op and empty products are the point.)_

In any case I think it should still be fine if we can find a different reason that |C| ~ *. What about: can we find conditions under which 1_C is initial in C? Is this for example true when C is unital as an operad?

I think can't be true that |C| ~ * for all symmetric monoidal infinity-categories or there wouldn't be any non-trivial elements in CMon(An).

But it should be true for when C^\otimes is symmetric monoidal and unital: Being unital implies Hom^act(emptyset, x) ~ for all x, and being symmetric monoidal (hence a cocartesian fibration) means that Hom^act(emptyset, x) ~ Hom_C(1_C, x). Moreover, |C| ~ is true in all cases relevant for Nikolaus' stabilization of operads theorem, since the symmetric monoidal infty-categories in question all have inital or terminal objects (be they the tensor unit or not).

And thanks! Your argument for |C| ~ * really pinpoints where my problem is.

FlorianAdler / AlgebraBonn

Comments K-theory notes Chapter I #6