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 ...

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, ...

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, $\...

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 ...

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 $...

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
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.

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 ...

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 ...

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 ...

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 ...

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.

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