All Questions
Tagged with homotopy-theory stable-homotopy
261
questions
7
votes
1
answer
560
views
What is the closure of the Eilenberg-MacLane spectra under limits? under colimits?
Every bounded spectrum is in the closure of the Eilenberg MacLane spectra under finite co/limits. Thus every bounded below (resp. above) spectrum is in the closure of the EM spectra under limits (resp....
- 62.6k
2
votes
0
answers
128
views
A basic computation with spectra
Let $\mathbb{E}=\big(E_n, \sigma_n\colon T\wedge E_n\to E_{n+1}\big)_{n\in\mathbb{N}}$ be a $T$-spectrum, either in the topological setting (with $T=S^1$) or in the algebraic setting (with $T=\mathbb{...
- 2,811
1
vote
1
answer
132
views
Can a finite, type $n+k$ spectrum be a (non-iterated) colimit of finite, type $n$ spectra for $k \geq 2$?
By the thick subcategory theorem, if $X, Y$ are finite $p$-local spectra of type $m,n$ respectively, then $Y$ can be built from $Y$ in a finite number of "steps" iff $n \geq m$. Here, a &...
- 62.6k
3
votes
0
answers
108
views
Does $K(n)$ detect minimal $K(n)$-local cell structures?
Let $X$ be a finite spectrum, and let $N = dim_{\mathbb F_p} H_\ast(X;\mathbb F_p)$. I believe that $p$-completion $X^\wedge_p$ may be built as an $N$-cell complex where the cells are shifts of the $p$...
17
votes
2
answers
719
views
For which $n$ does there exist a closed manifold of (chromatic) type $n$?
Let $p$ be a prime and $n \in \mathbb N$. Does there exist a closed manifold which is of type $n$ after $p$-localization?
When $n= 0$ the answer is yes. When $p = 2$ and $n = 1$ we can take $\mathbb R ...
- 62.6k
4
votes
1
answer
253
views
When is a thick subcategory the preimage of a weak Serre class under a homological functor?
Let $\pi : \mathcal T \to \mathcal A$ be a homological functor from a stable / triangulated category to an abelian category, and let $\mathcal C \subseteq \mathcal A$ be a weak Serre subcategory. Let $...
- 62.6k
5
votes
1
answer
285
views
Ring spectra structures on a certain spectral analogue of $\mathbb{Z}/2$
We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors
\begin{align*}
\mathsf{Forget} &\colon \mathsf{Ab} \to \mathsf{Sets},\\
\mathrm{Inv}...
- 11.5k
2
votes
1
answer
259
views
A question about cofiber diagrams in stable $\infty$-categories
My question is as follows say I have a commutative diagram
$\require{AMScd}$
\begin{CD}
X @>f>> Y @>g>> Z\\
@V \alpha V V @VV \beta V @VV \gamma V\\
X’ @>>f’> Y @>>g’&...
- 103
26
votes
1
answer
820
views
Are complex-oriented ring spectra determined by their formal group law?
To every complex-oriented ring spectrum $E$ there is associated a formal group law, which is a power series $F_E(x,y)\in E_*[[x,y]]$.
Suppose $E$ and $F$ are two complex-oriented ring spectra and ...
- 2,012
5
votes
0
answers
118
views
Variations on Thomason's equivalence between connective spectra and symmetric monoidal categories
There's a number of results relating monoidal categories to connective spectra (which are themselves equivalent to $\mathbb{E}_{\infty}$-spaces):
Symmetric monoidal categories model all connective ...
- 11.5k
6
votes
1
answer
274
views
Lewis's convenience argument for $\mathbb{E}_{\infty}$-spaces
The 1991 paper of Lewis, “Is there a convenient category of spectra?” proved that it is impossible to have a point-set model for spectra satisfying the following criteria:
There is a symmetric ...
- 11.5k
5
votes
1
answer
474
views
Categorical models for truncations of the sphere spectrum
Picard $n$-groupoids are expected to model stable homotopy $n$-types. So far this has been proved for $n=1$ in
Niles Johnson, Angélica M. Osorno, Modeling stable one-types. Theory Appl. Categ. 26 (...
- 11.5k
31
votes
0
answers
849
views
The central insight in the proof of the existence of a class of Kervaire invariant one in dimension 126
I understand from a helpful earlier MO question that the techniques leading to the celebrated resolution of the Kervaire invariant one problem in the other candidate dimensions yield no insight on ...
- 2,984
3
votes
0
answers
178
views
For which categories $D$ is a $D^{\vartriangleleft\vartriangleright}$-shaped diagram in a stable $\infty$-category a limit iff it is a colimit?
Throughout, I'll omit the "$\infty$" from the term "$\infty$-category".
It is well-known (and sometimes even included in the definition, although not by Lurie) that pushouts and ...
- 728
21
votes
1
answer
2k
views
Why does elliptic cohomology fail to be unique up to contractible choice?
It is often stated that the derived moduli stack of oriented elliptic curves $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ is the unique lift of the classical moduli stack of elliptic curves satisfying some ...
- 423