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Aug
17
comment Topological Boundary Map
Here's something that might be helpful for context. Given a (homotopy) cofiber sequence $A \to X \to X/A$ (say of based spaces), you can "rotate" this to a new cofiber sequence $X \to X/A \to \Sigma A$, and this procedure can be iterated indefinitely. An immediate consequence of this Corollary is that all the resulting long exact sequences in $E$-homology are all canonically isomorphic (up to degree shifts -- things get skewed once per rotation).
Aug
16
comment Why steenrod commute with transgression
Hi @rj7k8 -- it's been ages since I've thought about this stuff (and it looks like my reference to pp. 106-7 of May's Concise Course were actually off, at least compared to the edition I'm looking at now). But I would expect that this might come from the cofiber sequences defining the attaching maps for your CW-complex (along with an understanding of the behavior of your chosen homology theory on spheres and disks). If you give me a more precise reference to what you're confused about, I'd be happy to see if I can say more.
Aug
14
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Jun
8
comment What are $E_\infty$-rings?
@user43687 Ah, okay. Well, either Lurie's thesis or, in a slightly different direction, his ICM address (which discusses deformation theory in the derived context) would be a great place to start in that case.
Jun
8
comment What are $E_\infty$-rings?
@user43687 I don't think I've ever really written anything down about operads. My own understanding of them has come bit by bit, cobbled together from a number of different contexts. Can you be more specific about something that you don't yet understand but would like to?
Mar
27
comment What's the point of spectra?
@DmitriPavlov Interesting, thanks for pointing that out!
Mar
4
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Jan
31
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Jan
12
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Oct
31
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Sep
30
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Sep
27
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Sep
13
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Aug
20
comment What are $E_\infty$-rings?
@user40276 -- Right, exactly. (Although you'll actually want (at least) two total of suspensions on your source or loops on your target to deduce that this hom-set is an abelian group.)
Aug
18
comment What are $E_\infty$-rings?
@user40276 Ah, you're pointing out that spectra are all abelian ($\infty$-)group objects? Yes, certainly "$E_\infty$-algebras" in an arbitrary setting needn't be ring-objects in any sense (take the simplest example: sets!), but I think the term "$E_\infty$-ring" only refers to "$E_\infty$-ring spectrum".
Aug
16
comment What are $E_\infty$-rings?
@user40276 -- sorry, I don't quite understand your comment. Could you elaborate? $E_\infty$-rings are commutative monoids in Spectra under the smash product, just as commutative rings are commutative monoids in AbGrp under the tensor product.
Aug
2
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Jul
15
revised What are $E_\infty$-rings?
addendum on dg-algebras, simplicial commutative rings, and E_\infty-ring spectra
Jul
2
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