The homotopy-theory tag has no wiki summary.

**17**

votes

**1**answer

873 views

### The cell structure of Thom spectra

I would like to understand the cell structure of integrally oriented Thom spectra. A Thom spectrum over a space $X$ is something you can build from a stable spherical bundle, which is classified by a ...

**5**

votes

**1**answer

197 views

### Pontrjagin ring structure on homology of Eilenberg-Mac Lane spaces

Is there any good reference for the Pontrjagin ring structure on
$$
H_\ast(K(\mathbb{Z}/2,k);\mathbb{Z}/2)\cong H_\ast(\Omega K(\mathbb{Z}/2,k+1);\mathbb{Z}/2)?
$$
I am familiar with Serre's theorem ...

**3**

votes

**0**answers

185 views

### Are there CW structures on homotopy limits of CW maps?

Consider the diagram of finite CW complexes $X \stackrel{f}\leftarrow Y$ where $f$ is a cellular map and note that its homotopy colimit is precisely the mapping cylinder
$$C_H = \frac{X \sqcup (Y ...

**6**

votes

**2**answers

231 views

### Components of a loop space, semidirect products, and multiplicativity

Let $(X, x_0)$ be a based topological space, and $\Omega X$ its based loop space. The group of path components of $\Omega X$ is $\pi_0(\Omega X) = \pi_1(X, x_0)$. For brevity, let's call this group ...

**4**

votes

**0**answers

171 views

### Adding morphisms to a category without changing homotopy type

I have a really tame category $C$: there are only finitely many objects $C_0$, each hom-set $C(x,y)$ for $x,y \in C_0$ has at most one element, and (aside from identity morphisms) if $C(x,y)$ is ...

**2**

votes

**0**answers

111 views

### Twisted product structure on product of Eilenberg-MacLane spaces

The product of Eilenberg-MacLane spaces $K(\mathbb{Z}/2,1) \times K(\mathbb{Z}/2,2)$ is an $H$-space with respect to a "twisted product structure", where you add the product of the first entries to ...

**4**

votes

**3**answers

683 views

### When are maps between topological spaces homotopic?

I wanted to ask if there is any known mehod to quantify 'how many' homotopy classes of maps there are between two given topological spaces $X$, $Y$ (CW-complexes, say).
So far I had the following ...

**5**

votes

**0**answers

133 views

### Refined cofinality theorem for homotopy limits of spaces

If $C$ is a simplicial model category, $F\colon I\longrightarrow J$ a functor between small categories, and $X\colon J\longrightarrow C$ a diagram, the canonical map $holim_JX\longrightarrow ...

**4**

votes

**0**answers

219 views

### Does abstract nonsense of model categories determine the “nonlinear” morphisms of $L_\infty$ algebras?

Recall that a Lie algebra is a module for the operad $\mathrm{Lie}$, which is freely generated by a binary antisymmetric operation $\beta$ modulo an equation that is quadratic in $\beta$. There is a ...

**13**

votes

**3**answers

561 views

### What are finite homotopy types?

Starting from an ordinary 1-categorical point of view, there are various obvious candidate definitions for ‘finite homotopy type’:
The homotopy type of a simplicial set that has only finitely many ...

**32**

votes

**9**answers

2k views

### Homotopy as a general organizing principle

One of the realizations that led to the development of Homotopy Type Theory (HoTT) is that the ideas of homotopy theory have very broad applicability in mathematics. Indeed, Quillen model categories ...

**3**

votes

**1**answer

184 views

### Bousfield localization before and after taking homotopy

The ncatlab says:
Under suitable conditions it should be true that for $C$ a model category whose homotopy category $\mathrm{Ho}(C)$ is a triangulated category the homotopy category of a left ...

**19**

votes

**2**answers

1k views

### How does it End?

A recent project has forced my colleague and me to take a rather abstract approach to dynamical systems, and the following definition arose naturally in that context.
Let $\mathcal{C}$ be a category. ...

**6**

votes

**1**answer

242 views

### Identifying the little disk operad with parenthesized braids

Let $D_2$ be the topological operad of little disks. This operad can be modelled "combinatorially" in terms of an operad of groupoids called $\newcommand{\PaB}{\mathbf{PaB}}\PaB$, the operad of ...

**8**

votes

**1**answer

469 views

### Etale cohomology of and forms of algebraic groups

Let $k$ be a field, and $K$ its separable closure. Consider two different $k$-schemes, $X$ and $Y$, which become isomorphic upon extension of scalars to $K$: $X_K \cong Y_K$. Then the etale ...

**8**

votes

**1**answer

306 views

### Unicity up to homotopy of simplicial enrichments

On the one hand, in their paper Simplicial structures on model categories and functors, Rezk, Schwede and Shipley proved that a simplicial model category structure on a given model category is unique ...

**5**

votes

**2**answers

239 views

### Why is the path fibration a strong Hurewicz fibration?

In May and Sigurdsson "Parametrized homotopy theory" there is a general treatment of Hurewicz style model structures in Chapter 4, see definitions 4.2.1 and 4.2.2. I am trying to adapt these to a more ...

**12**

votes

**1**answer

319 views

### When does localization preserve homotopy type of classifying spaces?

Let $\mathcal{C}$ be a small category and $\Sigma$ a collection of morphisms in $\mathcal{C}$. Denote by $F_\Sigma:\mathcal{C} \to \mathcal{C}[\Sigma^{-1}]$ the usual quotient functor from ...

**1**

vote

**3**answers

408 views

### prequantization on $TM \bigoplus T^*M$

Let $M$ be a pre-symplectic manifold.In recent years several Geometrists are working on $TM \bigoplus T^*M$ which has fascinated the complex and Poisson geometry. In recent decade also Nigel Hitchin ...

**18**

votes

**2**answers

2k views

### Open Problems in Algebraic Topology and Homotopy Theory

Some time ago (I see it was initially written before 1999?) Mark Hovey assembled a list of open problems in algebraic topology. The list can be found here. Some of the problems I know about have been ...

**11**

votes

**3**answers

617 views

### Strictly commutative elements of $E_\infty$-spaces

Let $X$ be an $E_\infty$-space (not necessarily grouplike). Let $x \in \pi_0 X$ be an element; say that $x$ is strictly commutative if there is a map of $E_\infty$-spaces $\mathbb{Z}_{\geq 0} \to X$ ...

**5**

votes

**0**answers

172 views

### Motivic homotopy spectral sequence

I would like to have a question about the re-index convention. Let us consider a spectrum $E$ (I am mainly interested in motivic setting, however let's consider the simplicial case firstly, i.e. $E$ ...

**5**

votes

**2**answers

585 views

### Homotopy limit-colimit diagrams in stable model categories

It is shown in Remark 7.1.12 of (a newer version of) Mark Hovey's book Model Categories that, in a stable model category, homotopy pullback squares coincide with homotopy pushout squares. The argument ...

**8**

votes

**2**answers

346 views

### Correspondence between operads and $\infty$-operads with one object

Given a simplicial operad one can form its category of operators. This is a simplicial category with a functor to the category of finite pointed sets which is a bijection on objects and whose ...

**5**

votes

**2**answers

278 views

### How does one Segal-subdivide a 2-category?

Let $\mathcal{C}$ be a small category. Then, its Segal subdivision $\text{sd }\mathcal{C}$ is a new category whose objects are morphisms of $\mathcal{C}$, and a morphism from $f:x \to y$ to $g: w \to ...

**3**

votes

**0**answers

129 views

### Where does the ''hyper'' go?

There is a model structure on the category $Grpd$ of groupoids such that weak equivalences are equivalences of categories, cofibrations are the injections on objects (see nlab) and there is a Quillen ...

**2**

votes

**0**answers

69 views

### ideals in Linfty algebras

Where is anything written about ideals in $L_\infty$ algebras?
What is the best formulation either in terms of coderivation formalism or
explicit multivariable formulas. Does the `obvious' theorem ...

**8**

votes

**1**answer

419 views

### Three questions on $\operatorname{hocolim}$

I posted this on math.stack.exchange but didn't get a helpful response, so please let me try it here.
Let $D$ be a small category and $F:D\to sSets$ a functor.
There is a bisimplicial set indicated ...

**4**

votes

**4**answers

478 views

### What is the homotopy fiber of a fold map?

If $X$ and $Y$ are based spaces, let $p_X: X\vee Y\to X$ be the fold map, or projection, onto $X$.
What is the homotopy fiber $F$ of $p_X$?
I think I have an argument that $F$ is the half-smash ...

**19**

votes

**3**answers

617 views

### Is the counit of geometric realization a Serre fibration?

Recall that a Serre fibration between topological spaces is a map which has the homotopy lifting property (HLP) for all CW complexes (equivalently for all disks $D^k$). The Serre fibrations are the ...

**16**

votes

**3**answers

1k views

### Is there a scheme corresponding to the unit interval?

Can someone complete the following table?
$\begin{array}{cc} \text{Topology over } \mathbb{R} & \text{Topology over } \mathbb{C} & \text{Algebraic Geometry} \\\\ \hline \mathbb{R} & ...

**5**

votes

**0**answers

214 views

### Embedding tower in low codimension

If $F$ is a suitably nice functor from manifolds to spaces, it has a degree $k$ "polynomial" approximation $T_k F$ in the sense of embedding calculus. We set $T_\infty F := \mathrm{holim} T_k F$.
The ...

**6**

votes

**2**answers

395 views

### A category with weak equivalences which is not a model category

I'm only considering complete and cocomplete categories. A pair $(\mathfrak{X} , \mathfrak{W}) $ is, by definition, a category with weak equivalences if $ \mathfrak{X} $ is a category and $ ...

**8**

votes

**4**answers

742 views

### Algebraic K-theory and Homotopy Sheaves

Recently, when I was reading the definition of higher algebraic K-theory, I tried to give myself some motivation by looking at derived algebraic geometry. The constructions for algebraic K-theory ...

**2**

votes

**0**answers

228 views

### homotopy pullback/pushout

Is it true that homotopy pullback and homotopy pushout coincide in the category of spectrum? I
had a feeling that this is the case, but don't know where to find a proof or how to prove it.
Thanks!

**6**

votes

**5**answers

580 views

### What is a good basic reference on model categories?

I am looking for a general reference text on model categories, that contains all the basic results and definitions. I'm perfectly happy to be pointed towards a textbook, and I'm not looking for ...

**3**

votes

**0**answers

151 views

### Can the various proofs of the connectivity in the Blakers-Massey theorem be algebraicised?

This is related to Transversality in the proof of the Blakers-Massey Theorem. Is it necessary?, and other questions, such as the intuition behind the Freudenthal suspension theorem.
The answers and ...

**4**

votes

**1**answer

358 views

### Borel constructions, equivariant cohomology, and homotopy quotients of monoid actions.

Let $M$ be a (discrete) monoid acting on a space $X.$ We may take the quotient of $X,$ by this action, $X/M$ that is the coequalizer of the action map $M \times X \to X$ against the projection map. ...

**12**

votes

**1**answer

314 views

### Does the signature admit a homotopy coherent refinement?

Cobordism genera can often be refined to $E_\infty$-orientations in the sense of Ando-Blumberg-Gepner-Hopkins-Rezk:
1) the mod 2 Euler characteristic $MO\to H\mathbb{F}_2$;
2) the $\widehat A$-genus ...

**2**

votes

**1**answer

575 views

### What is the 31th homotopy group of the 2 - sphere ?

What is $\pi_{31}(S^2)$, the 31th homotopy group of the 2 - sphere ?
This question has a physics motivation:
1) There are relations between (2nd and 3rd) Hopf fibrations and (2- and 3-) qbits ...

**7**

votes

**1**answer

451 views

### Grothendieck fibrations and classifying spaces

Suppose that $$F:\mathcal{D} \to \mathcal{C}$$ is a Grothendieck fibration of small categories, whose fibers are groupoids. Is there anything sensible which can be said about the induced morphism ...

**4**

votes

**1**answer

354 views

### “monotone” homotopy?

This is a question about a concept that I call "monotone homotopy" which arises in a natural way in some topological situations.
Let $X$ be a (bounded) metric space, $Y$ be a topological space and ...

**1**

vote

**1**answer

269 views

### A computation by the Shapiro Lemma

Hi:
When I read the book "An introduction to Homological algebra" by Weibel, the page 206, line 9 says that
"Shapiro's Lemma tell us that
$H_q(S_n(X)\otimes_{Z}A)$ is zero if $q\neq 0$ and is ...

**0**

votes

**0**answers

144 views

### Change the fiber of a fibration

Let $F \rightarrow E \rightarrow B$ be a (Serre) fibration of topological spaces. Given a map $F' \rightarrow F$, is there a criterion for the existence or even an explicit construction of a fibration ...

**2**

votes

**0**answers

130 views

### Explicit Lie May structure on cosimplicial DG Lie algebras

In the paper "Homotopy Lie algebras", Schechtman and Hinich proved that any cosimplicial
differential graded Lie algebra has the structure of a 'Lie May algebra'.
If my understanding is right here, ...

**1**

vote

**1**answer

184 views

### Notation of a pregallery

I'm transcribing parts of Harm van der Lek's thesis 'The homotopy type of complex hyperplane complements' and due to it being written in 1983 the typesetting isn't very detailed. In latex, how should ...

**3**

votes

**1**answer

352 views

### Why $\Omega X$ and $BG$ are adjoint functors?

This is definitely not a research level question. I believe this is "common sense" among homotopists, however after "extensive" googling for 2 days I could not find a proof of it online or in standard ...

**16**

votes

**3**answers

601 views

### Visualize Fourth Homotopy Group of $S^2$

I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...

**5**

votes

**2**answers

1k views

### Generalized Categories for “Higher Homotopy Groupoids”

I was thinking about the definition of higher homotopy groups $\pi_n$ of a topological space in comparison to the common extremely formal fundamental groupoid construction of $\pi_1$. I'd like to be ...

**5**

votes

**1**answer

241 views

### What is the homotopy fiber of the map from a space to its James construction?

Given a pointed space X, let $J(X)$ denote its James construction. There is a natural inclusion $X\rightarrow J(X)$ which can equivalently be described as the unit map $\eta_X:X\rightarrow \Omega ...