24 votes
Accepted

Why do we study complex orientable cohomology theories

There is a sort of a priori reason why one would consider the cohomology theory $MU$, without first knowing of its connection to manifold geometry, to formal groups, … . Since complex-oriented ...
22 votes

Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

The question used the phrase "still needed." This is a very loaded term, and the answer that you give will depend very strongly on how you interpret it. If, as Dylan does, we interpret this as asking ...
21 votes
Accepted

Is $[X, \_]$ a homology theory?

This holds only for compact objects (i.e. finite CW spectra), since it is easy to see that additivity fails otherwise (the other axioms of homology theories are satisfied). The usual way to obtain a ...
  • 6,164
19 votes
Accepted

Morava $K(n)$'s are not $E_{\infty}$

The assertion is Lemma 5.6.4 in Rognes's "Galois extensions of structured ring spectra" available on the arXiv. In fact, the $K(n)$ spectra do not even admit $E_2$-algebra structures. The reason is ...
  • 24.6k
17 votes

Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

At the risk of starting some kind of (un?)civil war, let me expand on my comments. First and foremost, let's address the interpretation of the question. The OP asks "do we need a model category of ...
16 votes

Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

The original question has been answered in the sense that there are people who are confident to prove every statement about spectra they care about without recourse to models or model categories of ...
15 votes
Accepted

Do there exist "topologically significant" (and not "algebraic") triangulated categories killed by the multiplication by $p$?

Depending on exactly what you mean by "killed by $p$", the answer may be no. Let $\mathcal{C}$ be a stable $\infty$-category and let $\iota_{\mathcal{C}}$ be the identity functor from $\mathcal{C}$ to ...
13 votes
Accepted

Equivalent definitions of Thom spectra

The details of the comparison are treated in detail in the original ABGHR paper (and then unfortunately split in half across two papers in the updated version), so I'll just try to give a sketch of ...
  • 12.8k
13 votes

Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

Given that Sp is better behaved than all other existing models of spectra No, Sp is not better behaved than other models. The reason that it seems to be is because all operations in Sp (e.g., Ω^∞, Σ^∞...
11 votes

Do there exist "topologically significant" (and not "algebraic") triangulated categories killed by the multiplication by $p$?

Yes. For $p$ a prime and $n > 0$, the Morava $K$-theories $K(n)$ and their connective versions $k(n)$ are associative (but not commutative) ring spectra with coefficient rings $\Bbb F_p[v_n^{\pm 1}]...
  • 48.1k
11 votes
Accepted

Is the "inverse" (i.e., the "cohomological") numeration for singular (i.e., $H\mathbb{Z}$-)homology of spectra "acceptable"?

I'd say it is an unfortunate accident you did that once, and you should not do it again. The question is not mathematics but readability: not a good idea to go against a universally accepted ...
  • 29.2k
11 votes
Accepted

Connective spectra and infinite loop spaces

Ok, this discussion has grown beyond the level of comments so I'll collect the facts here. A bit of terminology: a $(-1)$-connected space is a space with a choice of basepoint and the category of $(-1)...
10 votes
Accepted

Higher coherent multiplicative structures on S-algebras

You can pick a model for the operad $E_n$ which receives a map from the associative operad. For instance, the Boardman-Vogt tensor product of the associative operad with $E_{n-1}$ has this property. ...
10 votes

Multiplicative Structures on Moore Spectra

Alright, I will be gutsy and try to provide the idea behind getting bounds for higher associativity of $M(p^i)$. I am taking the risk of prematurely displaying part of the work in my thesis in public ...
  • 1,983
10 votes

Multiplicative Structures on Moore Spectra

There is an unpublished result of Hopkins that none of the Moore spectra (modulo any power of $p$) admit $A_\infty$-structures. Although I do not know the proof, I believe the obstruction is $L_1$-...
  • 24.6k
10 votes
Accepted

Homotopy groups of $K(n)$-localization of the Brown-Peterson spectrum

See Lemma 2.3 of the following paper, and the surrounding discussion: ...
9 votes

Multiplicative Structures on Moore Spectra

I think the answer to your question is essentially unknown. As far as I'm aware the best known results are: $M(p)$ admits an $A_{p-1}$ structure but never an $A_p$-structure. I learnt the following ...
  • 3,595
8 votes

Definition of an E-infinity algebra

A completely explicit definition that works over any ring is given as Proposition 18 in the following preprint of Malte Dehling and Bruno Vallette that was posted on the arXiv today. http://arxiv.org/...
  • 37.5k
8 votes

Why do we study complex orientable cohomology theories

A surface-level motivation for complex oriented cohomology theories is that they are precisely those that admit a theory of generalized chern classes. More surprisingly, the universal complex ...
  • 1,230
8 votes
Accepted

Inverting objects in a symmetric monoidal category

To be clear, this claim refers to a very specific construction of $\mathcal{C}[X^{-1}]$, where you copy the construction of the localization of the ring and defines it as the colimits of: $$ \mathcal{...
  • 33.5k
7 votes
Accepted

Intuition - difference between Moore spectrum and Eilenberg-Mac Lane spectrum

I'm not sure I can answer to the general question, but I can explain why the Moore spectrum $\mathbb{S}/p$ shows up in the discussion of Bousfield localizations. This is just the cofiber of ...
  • 15.7k
7 votes
Accepted

When is the Thom spectrum of a virtual vector bundle effective?

Yes. A bit more generally, if $\xi$ is a perfect complex of rank $\geq 0$, then $Th(\xi)$ is effective (even very effective): the question is Nisnevich-local on $X$ and $\xi$ is locally a complex of ...
  • 7,964
7 votes
Accepted

Bousfield's distributive lattice DL and non-ring spectra

My paper A combinatorial model for the known Bousfield classes defines an complete ordered semiring $\mathcal{A}$ and a homomorphism from $\mathcal{A}$ to the Bousfield lattice mod the telescope ...
6 votes
Accepted

Basic questions on spectra

Here is a slightly more fleshed out version of the comment above. First, the claim that the collection $\mathcal{C} = \{ \Sigma^{p,q} U \mid U \in Sm/S, p,q \in \mathbb{Z} \}$ is a collection of ...
  • 3,595
6 votes
Accepted

Smash product of spheres in $\mathbf{SH}$ and product in cohomology

Thanks to Marc Hoyois which pointed the answer in a comment above. I am just writting that with more details: 1.- Yes, the diagram is commutative. 2.- This already happens in classic stable ...
  • 2,671
5 votes

Multiplicative Structures on Moore Spectra

The classical reference is as follows: ...
5 votes
Accepted

Does the (singular)cohomology of any acyclic spectrum vanish?

Let me address what hasn't been answered in comments (not in an optimal way, though). 1) is OK and, modulo the meaning of your quotation marks, the answer to 2) is 'no'. I mean, don't expect ...
5 votes
Accepted

Model category structure on spectra

Is there a model structure on Spt(S), having SH(S) as homotopy category, such that every object is fibrant? If so, could you provide a reference? No, if the given model category of spectra Spt(S) (...
5 votes

Model category structure on spectra

Since you tagged this as a reference request, let me give you some relevant references. The first injective model structure for motivic spectra that I am aware of is Jardine's paper Motivic Symmetric ...
  • 22.6k
5 votes

Stable Adams operations

We can define complex K-theory spectrum $K$ looking at even suspension maps $\Sigma^2:S^2\wedge BU\to BU$. The map $\Sigma^2$ has an expression as a classifying map for vector bundle $E\otimes \tau$, ...

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