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created some minimum at lim^1 and Milnor sequences
(sorry for the title, I am undecided as to which single singular term best to use here)
‘First derived limit and Milnor sequences’ perhaps.
I was trying to check the abstract characterisation since it was typed in wrong (was $lim$ instead of $lim^1$), and couldn’t line up the reference with my copy of Goerss+Jardine, so I edited the references to fit. Then I realised Urs is probably working from the freely available draft version captured by the Internet Archive. I haven’t re-edited it, but it’s worth a remark at Simplicial homotopy theory that the numbering is different, so I put a little warning there.
Thanks for fixing the typo.
Regarding the numbering: there is the edition of Goerss-Jardine in “Progress of Mathematics” and the one in “Modern Birkhäuser Classics”. I suppose I have been using both. They mostly agree, but from the former to the latter the section on spectral sequences of towers was moved from section VII.6 to section VI.2. That makes good sense, I suppose, since VI is “The homotopy theory of towers” while VII is “Cosimplicial spaces”. So I adjusted the numbering in the entry.
I was using the 1996 Progress in Mathematics version, and that’s what was referenced in the article at the time. I tweaked the warning note.
Okay, thanks.
That definition 2 of the Mittag-Leffler condition doesn’t look right. Should it be, for all $k$ there exists $i$ such that for all $j$… ?
Thanks, right. I have fixed it.
I have added statement and proof of the Milnor sequence for reduced generalized cohomology groups on CW-complexes (here)
I have also expanded the Definition section
I have spelled out the proof of the statement (here) that $\underset{\longleftarrow}{\lim}^1$ is indeed the first right derived functor of the limit functor.
I have also spelled out the proof of these properties of $lim^1$.
Hm, I should add why $Ab^{(\mathbb{N},\geq)}$ (towers of abelian groups) has enough injectives in the first place. Why?
Sledgehammer proof: if C is presentable so is Fun(K,C) for small categories K, and Grothendieck abelian categories have enough injectives.
Straightforward proof: Evaluation at n is exact and has both a left and right adjoint. The right adjoint will preserve injectives. Now every object injects into the product of these using the counit for each n.
Thanks!
(In the last sentence you mean the unit, not the counit.)
I have also hacked in a proof for the Milnor sequence for homotopy groups of a tower of fibrations (here).
(Could be beautified further, but I am out of time for the moment.)
I have added also the statement of the Milnor sequence for generalized cohomology of spectra, here.
Hmm, the appearance of $(-)_n \colon Ab^{(\mathbb{N}, \geq)} \to Ab$ with its adjoints puts me in mind of the 7 adjoints between $Ab^{\to} \to Ab$ I was discussing here in relation to the 5 of the Sierpinski topos.
So is there anything interesting to say about $\mathbf{H}^{(\mathbb{N}, \geq)}$, for an $(\infty, 1)$-topos, $\mathbf{H}$?
You once considered inducing from
$\{ \ast \to S^0 \} \hookrightarrow (\infty Grpd_{fin}^{\ast/})$an inclusion
$\mathbf{H}^{\Delta^1} \stackrel{\hookrightarrow}{\longleftarrow} [\infty Grpd_{fin}^{\ast/}, \mathbf{H}] = \mathbf{H}[X_\ast].$I guess $(\mathbb{N}, \geq)^{op}$ could be included in $(\infty Grpd_{fin}^{\ast/})$.
I have added (here) the following statement and its proof:
Given a cotower
$A_\bullet = (A_0 \overset{f_0}{\to} A _1 \overset{f_1}{\to} A_2 \to \cdots)$of abelian groups, then for every abelian group $B \in Ab$ there is a short exact sequence of the form
$0 \to \underset{\longleftarrow}{\lim}^1_n Hom(A_n, B) \longrightarrow Ext^1( \underset{\longrightarrow}{\lim}_n A_n, B ) \longrightarrow \underset{\longleftarrow}{\lim}_n Ext^1( A_n, B) \to 0 \,,$Guys, I am not an expert for this nor am aware of your grand plans, but to me this entry is not the real entry on derived limit functors, but rather a specialized entry on the first derived limit in the case of sequential limit and of the target category being the abelian groups with emphasis on the connection to stable homotopy theory and Milnor sequences. I think we may need two more entries, one on higher derived limits and one really about $lim^1$ but in general, including nonsequential case and the nonabelian case with the application in unstable homotopy. This is parallel but not belonging to MIlnor sequence realm. $lim^1$ is usually pronounced lim one what is good redirect for future general article on $lim^1$. Original paper of Whitehead defined $lim^1$ when the target category are nonabelian groups, as far as I recall.
The idea section says that the notation $lim^1$ is used only for sequential case, what is not true, even in classical literature like Gabriel-Zisman who take rather general small source category.
Wikipedia gives some interesting discussion on a wrong proof of Roos also relating to ML property and vanishing of lim one with values in abelian categories, error discovered after 4 decades of usage; the theorem is still true under an additional assumption on the target abelian category then in the original highly cited theorem. Vanishing of higher limits often depends on the additional axioms of set theory independent from ZFC.
I will now create an entry for lim^n and will wait for you about if we should have a separate page for general $lim^1$ or we have to cram already huge page on Milnor exact sequences with the nonabelian and nonsequential case.
I had felt the same as Zoran. The derived functors of Lim are important aspects of the cohomology of small categories and are much more general than just $lim^1$. In the non-abelian case they come into Bousfield and Kan’s discussion of homotopy limits and generally they relate to the (co)homology of holims.
In addition to what is written in the Wikipedia article note that Chris Jensen in his lecture notes (Les foncteurs dérivés de Lim et leurs applications en théorie de modules, SLN 254) showed that there were linearly ordered categories and rings such that all the $lim^n M$ were non-trivial for some system M of modules.
I inserted a related reference into strong homology entry.
Sensitivity of the results on vanishing of (higher) derived limit functors on additional axioms of set theory, and consequences on strong homology are discussed in
- Jeffrey Bergfalk, Strong homology, derived limits, and set theory, arXiv/1509.09267
this entry is not the real entry on derived limit functors
Propbably this is the reason why it is not titled “derived limit” but “lim^1 and Milnor sequences”.
I have tweaked the second sentence of the Idea-section to clarify the scope of this article. In its presents form, It’s a standard topic of any algebraic topology text, as you know.
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