New answers tagged homotopy-theory
4
votes
Accepted
Does a complex-oriented $E_1$ ring spectrum (not assumed to have graded-commutative homotopy groups) receive a map from $MU$?
The initial example of such an $E$ is the Thom spectrum $M\xi$ associated to the $E_1$-map $\Omega \Sigma BU(1) \to BU$, studied by Baker and Richter in "Quasisymmetric functions from a ...
- 8,352
6
votes
Two $E_\infty$ structures on infinite matrices
First note that if $f\colon\mathbb{R}^\infty\to\mathbb{R}^\infty$ is a linear isometric embedding and $u\in O(n)\leq O$ we can define $f_*(u)\in O$ by $f_*(u)(f(x))=f(u(x))$ when $x\in\mathbb{R}^n$ ...
- 54.6k
2
votes
On a generalized homotopy transfer theorem
The Homotopy Transfer Theorem (HTT) applies in great generality. Morally, any algebraic structure encoded by a cofibrant operad can be transferred along weak equivalences between bifibrant objects. ...
7
votes
Accepted
A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $
I'm not sure exactly what you were claiming, but here is a correct claim.
$\pi_1(N)$ has six order four elements $\{\pm i, \pm j, \pm k\}$, and there are three natural maps $p_1, p_2, p_3: N \to \Bbb{...
- 9,213
2
votes
A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $
I'm not too comfortable with the description of $N$ in terms of matrix tensor products, but the manifold $M/(\mathbb{Z}_2\times\mathbb{Z}_2)$ you describe in your comment is the manifold of pairs of ...
- 34.8k
25
votes
What are some toy models for the stable homotopy groups of spheres?
My favorite warmup example to the stable homotopy groups of spheres is the following differential graded algebra.
Let $A$ have the underlying ring
$$
\Bbb Z[y] \otimes \Lambda[x],
$$
a ring with a ...
- 50.5k
8
votes
What are some toy models for the stable homotopy groups of spheres?
As Dave Benson says, the Noetherian condition simplifies a lot of things. The derived category of a commutative ring satisfies many of the properties of the stable homotopy category. The derived ...
- 4,076
15
votes
What are some toy models for the stable homotopy groups of spheres?
You could say that I've made a living out of looking at the stable module category of a finite group (or rather its slight enlargement, the homotopy category of complexes of injective modules, $\...
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5
votes
Accepted
The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$
Let $R$ be a ring. $BGL(R)^+$ is homotopy equivalent to the $0$ component of $\Omega^\infty K(R)$, and it is stably equivalent to $BGL(R)$.
In particular, for a (co)homology theory $E$, understanding $...
- 12.5k
0
votes
Which positive flat stable model structures on (flavors of) spectra have the property that cofibrant operad-algebras forget to cofibrant spectra?
I asked this question over a year ago, when revising my paper (joint with Donald Yau) Smith Ideals of Operadic Algebras in Monoidal Model Categories. In the end, I thought this question was relevant ...
- 25.5k
1
vote
Reference for choosing a path lifting function?
(Not an answer, but long for a comment.) Spanier's "Algebraic Topology", Section 2.7, gives Hurewicz' proof of the theorem that a local (Hurewicz) fibration with respect to a numerable open ...
- 8,352
15
votes
Accepted
On the connections between condensed mathematics and homotopy theory
The way in which "condensed sets are similar to topological spaces" is very different from the way in which "$\infty$-groupoids are similar to topological spaces". In fact, ...
- 12.5k
2
votes
Reference for choosing a path lifting function?
Perhaps what you're referring to is Section 7.2 (page 57 of the pdf) of Peter May's A Concise Course in Algebraic Topology. Here he characterizes what it means to be a Hurewicz fibration in terms of ...
- 25.5k
4
votes
Accepted
On infinity-morphisms between algebras over algebraic operads
It is a typo. The map $f$ should only be assumed to be a morphism of the underlying graded $\mathbb S$-modules.
- 38.8k
3
votes
Accepted
Do finitely presentable $\infty$-groupoids precisely correspond to the finite cell complexes?
I answer the question "where can I read the formal definition of the presentation of ∞-categories by generators and relations?"
You can read about this in the Unicity paper by Barwick and ...
- 25.5k
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