All Questions
9
questions
4
votes
0
answers
440
views
An $E_{\infty}$-algebra is a $C_{\infty}$-algebra?
Past this question in MO have raised the following questions for me.
Question
In characteristic $0$, it is well-known that a Kadeishvili‘s $C_{\infty}$-algebra is an $E_{\infty}$-algebra.
However, do ...
- 869
5
votes
1
answer
209
views
Homotopy coherent localisation of a ring spectrum $E$ at a subset of $\pi_0E$
Homotopy coherent Invertibility.
Similarly to how $\mathbb{E}_k$-commutative spectra are a homotopy-coherent version of homotopy commutative spectra, encoding commutativity with higher homotopies, we ...
- 11.5k
4
votes
1
answer
387
views
The “field of fractions” of the sphere spectrum (localization at $\pi_0(\mathbb{S})\setminus\{0\}$, the non-zero integers)
Perhaps the most common construction of the rational numbers is the one given by taking the field of fractions $\mathrm{Frac}(\mathbb{Z})\cong\mathbb{Q}$ of the ring $\mathbb{Z}$ of integers.
I'm ...
- 11.5k
14
votes
1
answer
344
views
The first two $k$-invariants of $\mathrm{pic}(KU)$ and $\mathrm{pic}(KO)$
$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\pic{pic}$Real and complex topological $K$-theories, $KO$ and $KU$, have Picard spectra $\pic(KO)$ and $\pic(KU)$ built from the $\mathbb{E}_\infty$-...
- 10.3k
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
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
13
votes
4
answers
3k
views
Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?
This question asked whether $\mathrm{Sp}$ is convenient in the sense of satisfying (in the $\infty$-categorical sense) a list of desired properties of Lewis in his 1991 paper (see there).
The answer ...
18
votes
1
answer
2k
views
Is the $\infty$-category of spectra “convenient”?
A 1991 paper of Lewis, titled “Is there a convenient category of spectra?” proves that there is no category $\mathrm{Sp}$ satisfying the following desiderata$^1$:
There is a symmetric monoidal smash ...
- 11.5k
7
votes
0
answers
152
views
Stable category of pro-spaces
I am in the situation of having to understand groups $[\mathbb{S},E\wedge X_+]:=\pi_0\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)$ in $\text{Sp}(\text{Pro}(\mathcal{S}))$, ...
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