17
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 ...
- 50.4k
14
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
14
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, $\...
- 9,531
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
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
4
votes
What is the precise relationship between pyknoticity and cohesiveness?
Recent work by Qi Zhu, Fractured Structure on Condensed Anima, is looking to approach condensed mathematics via Jacob Lurie's concept of a fractured structure, seen as "local cohesive structure&...
- 4,901
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
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
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
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 ...
- 8,287
1
vote
Accepted
Reference request-Natural equivalence detected pointwise for complete Segal spaces
The assertion is stated and proved as Proposition 2.21 in arxiv.2311.01101.
- 1,617
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