## 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 ...

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 $x\in\mathbb{R}^n$ ...

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{...

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 ...

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 ...

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 ...

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, $\...

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 $...

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 ...

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 ...

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, ...

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 ...

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 ...

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