Comments K-theory notes Chapter II

Hi Ferdinand,

I know, it has been a very very long time since the end of the course, and you probably didn't expect a reaction anymore. I had started going through chapter II in detail, but then other things were going on and I forgot about it... I recently needed some of the stuff about \infty-operads and decided I should finish going through all of Chapter II, so here we are!

Below you'll again find a list of comments/typos/questions on your K-theory notes, this time on chapter II. The first part (maybe even up to three quarters of it) was typed in March/April this year. The rest earlier this month. I'm not quite sure whether they are of much use anymore, especially all the corrections of typos, since probably a lot less people are reading these notes now (though I'm sure they still serve as a reference for many people, they sure do for me! ^^) But maybe some of the remarks are still interesting.

Here we go:

p.60, remark after II.1: the remark sounds like that the justification for calling this the Segal condition is that it is a special instance of the Segal condition we have seen in I.64. To be fair to history, it would perhaps be good to point out that the true justification for calling this the Segal condition is that this condition first appeared in essentially the form of definition II.2 in Segals article Categories and Cohomology Theories, definition 1.2. A historically more accurate remark would perhaps turn your remark around and say that the Segal condition we saw earlier in I.64 is really a generalization of this Segal condition in II.2 and it is nowadays generally accepted to also refer to this general condition as the Segal condition.
p.61, II.2: I’ve had a think about this proposition and the remark that Fabian made before and after it. Here I like the perspective inspired by Emily Riehl better than the perspective of Fabian. She refers to Segal spaces without the completeness condition to ‘pre-\infty-categories’, see https://youtu.be/A6hXn6QCu0k?t=4508. I guess we could think about this as a category in which we do not necessarily demand that the isomorphisms are equivalent to the identities. In this language, a cartesian monoid in anima is really a pre-\infty-category with a single object, where the words ‘with a single object’ mean ‘the space of objects is contractible’. (As I said in the lecture at the time, and as you kinda also put in your notes, I really think we should only use the words ‘X is unique’ if the space of all choices of X is contractible, and not if it is only connected!) The question is now whether we should be disappointed that the statement is “a monoid in anima is a pre-\infty-category with a single object” rather than that "an \infty-category with a single object". A priori it seems that the latter is what we would expect from ordinary category theory. But is it? In ordinary category theory, we actually do not impose completeness on our categories! In fact, example I.68 indicates that 1-categories are rarely complete, only if the only isomorphisms are identities (like, on the nose identities.) The equivalences in the 1-category of sets are basically just much too strict to make completeness a useful notion. So the statement in 1-category land is really also ‘a monoid is a not-necessarily-complete category with a single object’, and thus maybe we should just be happy with that. But well, I haven’t talked with anyone about this opinion so maybe experts would disagree with me and say that “having a unique object” should actually mean that pi_0 vanishes… I’m curious what you / anyone else who reads this thinks about this.
p.62, II.2: In “A similar argument applies shows” there are two verbs.
Also, note that in the previous argument we only had functoriality in X, but that here we really need the functoriality in (X,x), which could perhaps be remarked.
p.62, II.2, bottom: in “where the colimit on the right-hand side is a good old colimit taken in sSet”, isn’t the colimit taken in Set?
btw, it’s absolutely great that you wrote down a proof of the equivalence!
p.63, top: in the first alignment, second line, shouldn’t it be the pullback of HomsAn(T,X) and {x} over Hom{sAn}(Delta^0,X), rather than of {const_x} over Hom_sAn(1_T,X)?
And then in the third line, I would expect the pullback to be of {x} over Hom_An(*,X_0).
p.64, II.5: you forgot a subscript “\geq 1” in the statement, around (*/Cat_infty)
p.68, proof sketch II.10: I guess we actually need that the adjunction of Sigma and Omega also works with (*/An)_{\geq 1} on the right.
p.71, near top: there is a typo: gj : yβ(j−1)+1 \otimes … \otimes z
p.71, bottom: there should be no ‘of’ in ‘the sub-groupoid of spanned by finite...’
p.74, top: just a tiny detail, but it seems that you write |GL_n(R)| for the underlying topological space before applying Sing, so that Sing(|GL_n(R)|) = GL_n(R) is really an equality, right? But at a later point you say it is an equivalence by the unit map…
p.75, bottom: why does it follow that Vect_R(X) ~ Hom_An(X,Vect_R)? To be completely honest with you, I cannot even figure out right now what a canonical comparison map would be between them. (I mean, given a vector bundle over X we should presumably send it to the functor sending x to the fiber over x, but it’s not clear to me what version of straightening I should apply to turn this into a map of animae.) And even if I had a map, it is not clear to me why this would induce equivalences on homotopy groups.
p.75, bottom: isn’t the argument for preserving E1-groups the same as that for E1-monoids? If you preserve products, then the shear map is also preserved, right?
p.82, diagram II.18b.2: I’m a little thrown off by the fact that you make it sound as if the crucial statement is the commutativity of the diagram, rather than the statement that the forgetful maps are equivalences. (At first I didn’t even notice you wrote these equivalence signs there.) Possibly I’m misunderstanding what both of the forgetful functors are, because to me it seems like the commutativity of this diagram is completely trivial: I thought they were both just pre-composing by Gamma x {[1]} → Gamma x Delta.
p.83, top: okay, yes, clearly I’m misunderstanding something, because I don’t see how the equality of the functors CMon(Mon(An)) → Mon(An) and Cut*: Cmon(Mon(An)) ~ CMon(An) → Mon(An) should follow ‘easily from (II.18b.2)’. To me it seems like one is given by precomposing with Delta x {[1]} → Gamma x {[1]} → Gamma x Delta while the other one is precomposing with {<1>} x Delta → Gamma x Delta.
p.84, bottom: finally it starts to become clear. The crucial diagram is the one at the bottom of page 84. It seems to me that diagram (II.18b.2) does not say what you are intending, namely that you are trying to express equivalence of the two ways of ending up in Mon(C), either through CMon(C) or directly. Could this be?
p.84, top: ‘solid’ diagram?
p.85, top: aren’t both maps left inverses of the unit map? Like, in both triangle identities we use the counit in the second map.
p.85, top: the 'can not' should be 'cannot'.
p.87, bottom: in the last diagram, one of the x c should be x d.
p.88, Free E\infty monoids: typo: the forgetful functor has a left adjoint.
p.91, II.23c: So I had a thought about this proof. In this case, N^op is of course very simple and in fact its 1-skeleton of N^op is final in N^op and is even simpler, so it follows immediately that we may test for equivalences 'pointwise' by assembling all the pointwise inverses by hand. But I expect that it is always true in limit categories that equivalences can be tested 'pointwise' (i.e. after applying each of the structure maps.) I think this should follow from the fact that any object X in a limit of some functor I -> Cat_\infty can be written as the colimit over all its projections to X_i in a functorial way.
p.92, example II.25(a): more accurate would perhaps be to say that Sp(C) ~ */C since we haven’t assumed that C is pointed. Or does this follow somehow?
p.94, bottom: it is really nitpicking, but the phrase ‘pushouts agree with pullbacks’ could perhaps more accurately be described by ‘pushout squares agree with pullback squares’ since with a ‘pushout’ we often mean the resulting object. (Compare with 'finite products agree with finite coproducts'.)
p.95, II.29a: very nice catch! And good explanation :)
p.96, II.30: “it also has a left adjoint given by...“ Somehow the appearance of the loops there feels really strange to me, I would have expected a suspension if we are ever going to expect this to give a stable \infty-category. What are your thoughts on this? Also, do you happen to have a reference for this statement?
p.101, top of the page: I guess it’s almost the same, but alternatively one could continue the calculation of C_*(S) by noting that 1) the diagonal is (co)final in {(n,i) | n >= i} and 2) \tilde{C} turns suspension into shifts, so that we obtain colimi \tilde{C}\*(S^i)[-i] = colimi \tilde{C}_*(S^0) = \tilde{C}_*(S^0) = Z[0] = C_*(*). This proves that C_*(S) = C\*(*) and the rest of the lemma now follows directly from your final sentence.
p.102, top of the page: in the second line of the calculation a Hom is missing.
p.103, II.34: forgot to introduce X as an object of An^fin.
p.103, II.34: Let me think out loud here and try to figure out why it is actually an inverse functor. My first instinct was: well we can write any cell complex as an iterated pushout of spheres, and pushouts are sent to pullbacks so we obtain the value at any finite cell complex from the values on the spheres. But this is of course nonsense, since it is covariant, so we should really say that pushout squares go to pullback squares. I guess what we should do is find a canonical comparison map between F and \Omega^\infty(- \otimes F(S)) and prove inductively on the number of cells that it induces equivalences on all finite pointed cell complexes. For the spheres this is automatic, and then for each additional cell we use that both sides are reduced and excisive so induce fiber sequences. We then know that the maps on fibers and on total spaces are equivalences by induction, and thus we get it also for the base. Does this make sense? I just don’t see how to construct a comparison map in either direction. Any ideas?
p.107, “By the Segal condition and the fact that has p-cocartesian lifts for all inert morphisms, it suffices to check that it has p-cocartesian lifts for the unique active maps fn : -> <1> (we leave it as an exercise to work out a proper argument).” This was indeed claimed by Fabian in the lecture, but turned out to be false, as Sil luckily observed. (Sil pointed out that if it were true, then the same reasoning would give that in fact cocartesian lifts for <2> → <1> would suffice. Interestingly, Fabian called it absurd that Sil would even propose this.) The rest of your discussion seems to be completely correct though, and is in fact a nice characterization!
p. 112, in the middle of the page, I think that in the domain/codomain of alpha*, the hom-spaces should be taken in P(E)/X rather than P(E/X). Also, I feel that here it’s slightly easier to argue that P(E)/X is generated under colimits from the objects coming from E/X, since this should follow from the fact that P(E) is generated under colimits by E and that colimits in P(E)/X are computed in P(E). So then alpha* is an equivalence on all objects and thus alpha is an equivalence.
p.112, perhaps it’s also noteworthy to say that there is a surprisingly short proof of II.38 in Ando, Blumberg, Gepner, Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map, section 5.3. (Though I have to admit I don’t get their argument for why the functor is fully faithful.)
p. 113, unfinished sentence around the second diagram.
Did we actually ever define maps of infty-operads?
p. 118, “opoosite” should read “opposite”.
p. 125, definition II.45, you probably want that Op^semi-add and Op^add are subcategories of Op^x and similarly Op^st is a subcategory of Op^lex?
p. 129, example II.51(b)(ii). I like your explicit calculation with the Day convolution a lot, since it’s insightful to see how Day convolution turns out in practice. But I think there’s actually a much easier way to see that the induced monoidal structure on pointed anima(e) agrees with the usual smash product, using the fact that we already know that the functor (-)_ An ---> An_ is strong monoidal. Indeed, any pointed space (X,x0: → X) can be written as the cofiber in pointed spaces of the map x0_+: S^0 = _+ ---> X_+, and similarly for Y, and since smash product commutes with colimits in both variables we can now compute it as a colimit of smash products of objects that come from An and those smash products we already know. You’d get a square with legs X_+ ← S^0 ----> Y_+ and X_+ ---> (X x Y)_+ ← Y_+. By computing the colimit of this square in two steps you see it’s the colimit of the map (X v Y)_+ ---> (X x Y)_+ and this is our desired formula for the smash product :)
p. 129, “brave new algebra”, the first sentence isn’t properly formulated.
p. 130, very top, why do we need to write it as a composition of two functors? Isn’t Sp also already the stabilization of An^x? So we could apply II.51 directly, right?
p. 131, end of proof II.53a), okay it’s a nice proof, but let’s be fair: we didn’t actually use our formula for the tensor product, right? What you basically did was: 1) show that colimit preserving functors out of spectra are determined by what they do on the sphere spectrum, and 2) argue that both sides preserve colimits in E’ and agree on the sphere spectrum.
p. 132, lemma II.52, you say that H: D(Z) → Sp is a left adjoint of the forgetful functor rather than a right adjoint.
p. 135, at some point you say “but now we need to show that π_∗ sends coproducts of spectra to products”, but the latter should read “to coproducts”, right? And then we also need to remark that finite products of abelian groups agree with finite coproducts.
p. 137, Theorem II.56a, I’m confused: what does it mean “inert in Delta (not in Delta^op)”? The inerts are precisely the ‘inclusions of intervals’, right? Which are equivalently those maps in Delta^op that are sent to inerts in Gamma^op? I don’t understand what difference the “op” would make.
p.140, Corollary II.59g, I suppose you mean that it is left adjoint to the forgetful functor, right?
p.146, II.62b, it should say ‘another N in degree 0’ rather than ‘another M’.
p.146, II.62b, are you sure you also want extra face maps and not only those extra degeneracies? I think having an augmented simplicial object with extra degeneracies already implies that the geometric realization of the underlying simplicial object is equivalent to the object at level -1, right? It looks strange that we’d need this cyclic construction, with A acting on the right on N, since I’d expect something similar should hold for bimodules if A isn’t commutative.
p. 147, II.62c, for the statement of part (a) that you prove at the very end, why do we need part (c) here?

FlorianAdler / AlgebraBonn

Comments K-theory notes Chapter II #10