**3**

votes

**1**answer

223 views

### Does the fat geometric realization take limits to homotopy limits?

I am deeply confused about geometric realizations and finite limits.
Suppose that I am working with simplicial sets (I dont need simplicial spaces) that are "good" in the sense of Segal so that I can ...

**11**

votes

**2**answers

256 views

### Quasicategories for non-simplicial model categories

If I have a simplicially enriched model category, then I can take the coherent nerve of the full subcategory of bifibrant opjects to obtain a quasicategory. If I have a model category that is not ...

**2**

votes

**1**answer

235 views

### coproduct of the homology of iterated loop space on spheres

Let $\Omega^{n+1}S^{n+1}$ be the base-pointed $(n+1)$-iterated loop space on the $(n+1)$-sphere. In the paper The homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, Lecture notes in ...

**5**

votes

**1**answer

485 views

### reference for “Topological algebra of Grothendieck”

I would like to have some references for Grothendieck's theory of "Topological algebra": a synthesis of homotopical and homological algebra,
with special emphasis on topoi.

**8**

votes

**0**answers

336 views

### Standard model structures on $Top$

Call a model structure on $Top$ (the category of topological spaces) standard, if the weak equivalences are the weak homotopy equivalences. In this nLab page, two standard model structures on $Top$ ...

**3**

votes

**1**answer

152 views

### Is a pullback along a Dold fibration a homotopy pullback?

Let $$
\begin{array}{ccc}
A & \to & B
\cr\downarrow&&\downarrow
\cr
A'& \to &B'
\end{array}
$$ be a pullback square in the category of all topological spaces (not just in a ...

**1**

vote

**1**answer

112 views

### Does a topological hypercover always have free degeneracies?

This question arises when I am reading Dugger and Isaksen's "Hypercovers in topology". According to Definition 4.1 in that paper, A hypercover of a space $X$ is an augmented simplicial space $U_*\to ...

**0**

votes

**0**answers

110 views

### reference for groupoid cohomology

In nLab (groupoid cohomology) says:
"Under the homotopy hypothesis theorem, plain (non-internal) groupoid cohomology is the same as the cohomology of homotopy 1-types."
Are there references for ...

**19**

votes

**1**answer

606 views

### Is there a generalization of homotopy groups to fractional dimensions

Does there exist a reasonable candidate for such an object as $\pi_{\frac12}(X)$?

**7**

votes

**1**answer

403 views

### Loop space generalization

Let $X$ be a based connected space. The space of based continuous morphisms $Top_{\ast}(S^1,X)$
is the space of loops $\Omega X$. Since $S^1$ is homotopy equivalent to the Eilenberg-Mac Lane Space ...

**3**

votes

**0**answers

236 views

### Quasicategorical Construction of a Cosimplicial Map of Rognes

In John Rognes' Galois theory monograph he constructs something called the Hopf-cobar complex for a coalgebra object $H$ (in spectra) and a comodule algebra $X$. It is, intuitively, the object whose ...

**0**

votes

**1**answer

332 views

### Dimension of two homotopy equivalent manifolds [closed]

Let $M,N$ be a closed (connected, without boundary, say smooth) manifolds which are homotopy equivalent. Does it follows that they are of the same dimension? One should be aware of examples of ...

**1**

vote

**0**answers

395 views

### Are there connections between Homotopy type theory and Grothendieck's theory of motives? [closed]

Are there any "visions" (maybe "dreams"), future plans or connections between Homotopy type theory and Grothendieck's theory of motives (or at least "connections" with universal cohomology theory)?

**2**

votes

**0**answers

66 views

### section spaces related to configuration spaces

In the paper Configuration spaces of positive and negative particles, Dusa McDuff, a section space $\Gamma(M)$ is constructed:
And in the paper ON THE HOMOLOGY OF CONFIGURATION SPACES. C.-F. ...

**1**

vote

**0**answers

105 views

### homotopy equivalence between configuration spaces on non-homeomorphic spaces

(1). Let $D^m$ be the closed $m$-disc in $\mathbb{R}^m$. For each $k$, does the $k$-th configuration space on $D^m$ homotopy equivalent to the $k$-th configuration space on $\mathbb{R}^m$
$$
...

**4**

votes

**1**answer

223 views

### Properties of loop space functor from homotopy types to group objects in homotopy types

I am trying to understand some properties of categories enriched in homotopy types, and the following question has become important:
When we take the loop-space of a (connected) homotopy type, we get ...

**4**

votes

**1**answer

191 views

### Is the simplicial set $(\Delta_3/\partial \Delta_3)^{\Delta_1}$ finite?

A simplicial set is called finite if it has only a finite number of non degenerate simplicies (or equivalently, if its number of $n$-simplicies grows polynomially with $n$). In an answer to a ...

**2**

votes

**0**answers

72 views

### cohomology ring of mapping spaces

In the lecture notes The homology of $\mathcal{C}_{n+1}$–spaces, n ≥ 0. F. Cohen, 1978, page 228-231, the cohomology ring
$$
H^*(\text{Map}_*(S^n, S^n\wedge X);\mathbb{Z}_p)
$$
is obtained for any ...

**2**

votes

**1**answer

129 views

### Configuration spaces of positive and negative particles

In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, page 3, line from bottom 1-3, it is given that for a $m$-manifold $M$, there is a map from the labelled ...

**4**

votes

**1**answer

193 views

### Stabilization of a generic pointed model category

Let $\mathcal C$ be a pointed model category. It is well-known that its homotopy category $\mathrm{Ho}(\mathcal C)$ is naturally a $\mathrm{Ho}(\underline{\mathrm{sSet}}_*)$-category, where ...

**0**

votes

**0**answers

446 views

### maps from labelled configuration space to section space / iterated loop space

In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, 2014, page 3, Section 3:
for a $m$-manifold $M$, consider the disc bundle $D(M)$ in the tangent bundle ...

**0**

votes

**1**answer

58 views

### Group completion of labelled configuration space on Euclidean spaces

In the lecture notes The Homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 225 -226, it is obtained that there is a group completion on homology
$$
\alpha_n: C(\mathbb{R}^n;X)\to ...

**2**

votes

**1**answer

163 views

### Equivalent definition of a Kan fibration

It is known (follows for example from Proposition 4.2 of Simplicial Homotopy Theory by Goerss and Jardine) that a Kan-fibration can be defined as a map having the right lifting property with respect ...

**4**

votes

**1**answer

288 views

### Natural transformations induce homotopies - Is this true in the “fat” world?

Let $\mathcal{C}, \mathcal{D}$ be categories internal to topological spaces (or compactly generated Hausdorff spaces, if you like) $F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ be continuous functors ...

**4**

votes

**0**answers

176 views

### Does the fat realization of simplicial spaces commute with finite limits up to homotopy?

I'm willing to work in the category of compactly generated Hausdorff spaces.
The ordinary geometric realization of a simplicial space commutes with taking pullbacks and preserves the terminal object, ...

**14**

votes

**2**answers

741 views

### Counterexamples for strengthening Whitehead's theorem?

Let $f:X\to Y$ be a pointed map of pointed connected $n$-dimensional CW complexes. Whitehead's theorem says that if $f_*:\pi_qX\to \pi_qY$ is an isomorphism for $q\le n$ and a surjection for $q=n+1$, ...

**4**

votes

**1**answer

451 views

### $\pi_8(S^5)=\pi_8(SO(6))=\mathbb{Z}/24$?

I am currently thinking about a physics model related to framed bordism $\Omega_3^{fr}=Z/24=\pi^s_3$, and the first stable example is $\pi_8(S^5)$, so I was curious about the generator, and happened ...

**3**

votes

**1**answer

123 views

### group completion theorem by using homology fibrations

In the paper Homology fibrations and group completion theorem, McDuff-Segal (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281:
Let $M$ be a topological monoid such that $\pi_0M$ is generated by ...

**5**

votes

**0**answers

166 views

### Moore spectra are not E-infinity (oldest known proof)

Fix a prime $p$. Let $M_p(i)$, the $i$-th Moore spectrum at the prime $p$, be the cofiber of the map
$$ S^0 \overset{p^i}\longrightarrow S^0 $$
where $S^0$ be the sphere spectrum. In the Mathoverflow ...

**1**

vote

**0**answers

81 views

### Internal Hom on simplicial presheaves and the preservation of cofibrant objects

1)Let $\mathcal{C}$ be a cartesian closed small category. Let $\operatorname{Map}\: : \: sPsh(\mathcal{C})\times sPsh(\mathcal{C})\to sPsh(\mathcal{C})$ be the internal Hom of simplicial presheaves, ...

**6**

votes

**2**answers

491 views

### On combinatorial and cellular model categories and infinity categories

I am looking for a counterexample. Let me first give the set-up. When you work with model categories, it is extremely common to assume they are cofibrantly generated. For me, this means the definition ...

**8**

votes

**0**answers

201 views

### Simplices and cubes

Question: What is the first appearance in the literature of one of the
following statements:
The result of intersecting a simplex with a cell of the dual
subdivision is a cube
There ...

**6**

votes

**3**answers

510 views

### Higher refinement of Seifert-van Kampen theorem on the language of hocolim

I like the following version of SvKT. If $\Pi_1$ is the functor of fundamental groupoid and $(X_i)_{i\in I}$ is a diagram of spaces then
$$\Pi_1({\sf hocolim}\: X_i)\simeq {\sf hocolim}\: ...

**4**

votes

**1**answer

176 views

### Does the CGWH-fication change the (weak) homotopy type?

Let $Top$, $CG$, $WH$, $CGWH$ be the categories of topological spaces, compactly generated spaces, weak Hausdorff spaces and compactly generated weak Hausdorff spaces.
There is the CG-ification ...

**7**

votes

**1**answer

144 views

### Are maps homotopic with respect to a uniform number of local homotopies

I've encountered the following problem that I'm sure someone more topologically inclined can answer:
Say that a homotopy of maps $f:X\times[0,1)\to Y$ between two compact smooth manifolds $X$ and $Y$ ...

**4**

votes

**1**answer

304 views

### $\Omega X$-action on spectral $X$-bundles

I can generalize the notion of a space over $X$ (where $X$ is a based and connected space) to the notion of a spectrum over $X$ by considering functors of quasicategories $X\to Mod_\mathbb{S}$. In the ...

**9**

votes

**1**answer

252 views

### Tensor products over operads and bar constructions

Let $O$ be an operad in spaces, $A$ an $O$-algebra and $R$ an right $O$-module. One can define $R \otimes_O A$ as the coequalizer of the two maps $ROA$ to $RA$. One can also define $B(R,O,A)$ (as in ...

**0**

votes

**3**answers

123 views

### smash product of pointed spaces preserve weak equivalences

Consider the category of pointed simplicial sets with usual notion of weak equivalence. The question is does the functor
$$Y \mapsto X\wedge Y$$ preserve weak equivalences? or at the very least does ...

**-2**

votes

**1**answer

143 views

### configuration space and iterated loop space

Let the topological monoid $M$ be the configuration space $C(\mathbb{R}^n;X)=C_n(X)$ as in the book The geometry of iterated loop spaces, Theorem 5.2. I want to prove that the map $\alpha_n$ in ...

**15**

votes

**0**answers

713 views

### Derived functors - homotopical vs homological approach

This question is a crosspost of the second part of this MSE question.
In my first course in homological algebra, derived functors were defined in terms of universal $\delta$-functors. In the text ...

**4**

votes

**1**answer

270 views

### localization and $E_{\infty}$-spaces

Let $\mathrm{Top}$ be the model category of topological spaces. Define a new model structure on $\mathrm{Top}$ where $f:X\rightarrow Y$ is a weak equivalence iff ...

**9**

votes

**2**answers

483 views

### Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections

I previously asked this on Math.SE but didn't receive a satisfactory answer.
Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: ...

**2**

votes

**0**answers

96 views

### Nilpotency of BG^+

I have become interested in knowing if there are conditions, in the literature, under which the action of the fundamental group of BG^+ acts nilpotently on its higher homotopy groups, where the + ...

**3**

votes

**1**answer

94 views

### Segal maps for Segal precategories

A Segal precategory is just a simplicial space $X:\Delta^{op} \to sSet$ such that its $0$-th space is discrete (i.e. constant). A Segal category is defined everywhere in the literature as a Segal ...

**8**

votes

**1**answer

187 views

### Direct proof that $U$ is an $E_\infty$-space

An immediate consequence of Bott periodicity is that the infinite unitary space is an infinite loop space and so an $E_\infty$-space. I wonder if there is a direct proof (not using $U = \Omega^2 U$) ...

**0**

votes

**1**answer

118 views

### iterated loop spaces and configuration spaces [closed]

In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map
$$
\phi: Hom_T(X,\Omega Y)\to Hom_T(SX,Y)
$$
is defined. And a map
$$
...

**9**

votes

**1**answer

296 views

### When does the free loop space fibration split?

This question is a repost from stack.exchange. It didn't get a lot of attention there. Perhaps it is badly written (or silly?). If so, I'd be happy to get comments/suggestions about that.
Let $X$ be ...

**5**

votes

**1**answer

206 views

### Map between homotopy groups of O, related to J-homomorphism and K-theory of Z

Let $s \geq 0$ be fixed. The $J$-homomorphism includes $\pi_{8s+1}(SO) = \mathbb Z/2$ in $\pi_{8s+1}^s$, the $(8s+1)$-th stable homotopy group of spheres.
Now regard $\pi_{8s+1}^s = \pi_{8s+1} ...

**33**

votes

**1**answer

887 views

### What can topological modular forms do for number theory?

Topological modular forms ($TMF$) have in the recent years made an impact in algebraic topology. Roughly, the spectrum $tmf$ is the (derived) global sections of the sheaf of $E_\infty$ ring spectra ...

**7**

votes

**0**answers

383 views

### What's the detailed proof of “the composition of planar tangles is well-defined”?

In the planar algebra theory (see here or there section 2), a planar tangle is an isotopy class; then to define the composition of two tangles, we need to choose a representative in each classes. See ...