Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas ...
1
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1answer
53 views
Two model structures of some category
Let $\mathscr{C}$ be a category and $(W,C,F)=$(weak equivalence,cofibration,fibration) is some model structure of $\mathscr{C}$. Another model structure of $\mathscr{C}$, $(W_{S},C,F_{S})$ are called ...
1
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0answers
19 views
Deformation sublevel sets of functions which preserve boundary
I'm interested in proving the following fact, which seems to naturally arise from gradient flow deformations, but appears to be a bit tricky.
Consider a smooth family
$$f_s : M \to \mathbb{R}, \quad ...
-5
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0answers
63 views
Are HOTT evaluated as alternatives to foundations of mathematics completely free from Gödel's incompleteness theorem?
I wonder how mathematicians who study Homotopy Type Theory think of this. Are they well aware of Gödel's incompleteness theorem?
4
votes
0answers
116 views
What does it mean to say the first Goodwillie derivative of $TC$ is $THH$?
A paradox:
Goodwillie calculus considers only finitary functors.
$TC$ isn't finitary.
Yet in some sense $\partial(TC) = \partial(K) = THH$ is the crux of the Dundas-Goodwillie-McCarthy theorem.
(...
10
votes
1answer
291 views
Homotopy orbits, spectra and infinite loop spaces
Let $X$ be an (naive) $O(n)$-spectrum (I'm choosing to work with orthogonal spectra). I've recently come across the following results,
$$(S^{n-1} \wedge X)_{hO(n)} \simeq X_{hO(n-1)}$$
and
$$\Omega^...
6
votes
0answers
98 views
Product-preserving fibrant replacement functor for the Joyal model structure
There are a few fibrant replacement functors for the Quillen model structure on simplicial sets that preserve finite cartesian products, namely $\operatorname{Ex}^\infty$ and $\operatorname{Sing}(|\...
12
votes
0answers
155 views
How many cells are needed in a simplicial structure of $\mathbb{S}^n$ to induce all of $\pi_n(\mathbb{S}^m)$
Serre proved, that for (allmost) all $n,m\in\mathbb{N}$ the homotopy groups $\pi_n(\mathbb{S}^m)$ are finite, so - using simplicial approximation - for $n, m$ fixed there is a finite cell ...
9
votes
3answers
328 views
Induced maps on homotopy groups by self maps of $\mathbb{CP}^n$
Let $f:\mathbb{S}^2\to \mathbb{S}^2$ with degree $d$.
It is well known that the induced map $$f_\ast:\pi_3(\mathbb{S}^2)=\mathbb{Z}\to \pi_3(\mathbb{S}^2)=\mathbb{Z}$$ is given by multiplication by $...
4
votes
0answers
124 views
Deforming a section to a section without zeros
Let $M$ be an oriented manifold of dimension $n$. Suppose furthermore that $E$ is an oriented vector bundle of rank $n-1$ over $M$. Let $s$ be a section of $E$ transversal to the zero section in $E$. ...
4
votes
0answers
86 views
Simplicial homotopy groups - reference request
I am looking for a reference for the definition 2.6 in https://ncatlab.org/nlab/show/simplicial+homotopy+group, which states
"The simplicial homotopy groups of any simplicial set, not necessarily Kan,...
2
votes
2answers
213 views
Basic questions on spectra
I have a basic question on Voevodsky's stable homotopy category of spectra $\mathbf{SH}(S)$, where $S$ is a finite dimensional noetherian scheme.
Let $E$ be an $\Omega$-spectrum and $\varphi \colon ...
4
votes
1answer
102 views
$X^K$ a Kan complex, without model structure or anodyne extensions
If $X$ is a Kan complex, then it is an easy consequence of the existence of the Quillen model structure or of the basic theory of anodyne extensions, that $X^{K}$ is also Kan. However, I am interested ...
1
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1answer
102 views
Why is the Chern Number Invariant under A Continuously Shrinking of the Structure Group?
In Witten's paper Three Dimensional Gravity Revisited and Quantization of Chern-Simons Theory with Complex Gauge Group, he used a fact that for a principal $G$-bundle, the quantization of the Chern ...
5
votes
0answers
55 views
When is a bisimplicial set diagonal fibrant
Let $sSet^2$ be the category of bisimplicial sets.
In the diagonal model structure on $sSet^2$ weak equivalences are diagonal weak equivalence (i.e.$ X \rightarrow Y$ is a weak equivalence if $dX \...
4
votes
0answers
82 views
Minimal infinity categories in the Segal space picture
There is a well-known notion of a minimal Kan complex (see Goerss/Jardine's book) which is generalized to a minimal quasi-category in these notes by Joyal
www.math.uchicago.edu/~may/IMA/Joyal.pdf
(...
3
votes
2answers
250 views
Quillen equivalence, fibrant objects
Suppose that $G:M\leftrightarrow N: U$ is a Quillen equivalence between two model categories. Suppose that $a\in M$ is a fibrant object, is it true that there exists always a fibrant object $b\in N$ ...
12
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0answers
363 views
What's wrong with the obvious argument that the unstable motivic category is an $\infty$-topos?
Fix a base scheme $S$, and let $Sm_S$ be the (ordinary) category of smooth schemes over $S$.
Denote by $Psh(Sm_S)$ the $\infty$-category of $\infty$-presheaves on $Sm_S$.
Let $Sh_{Nis}(Sm_S)\...
2
votes
1answer
184 views
Relationship between the Betti numbers $b_i(M;\mathbb{Q})$ and the dimension of rational homotopy $\dim_{\mathbb{Q}}\pi_i(M)\otimes\mathbb{Q}$
What is the relationship between the Betti numbers $b_i(M;\mathbb{Q})=rkH_i(M;\mathbb{Q})$ of a simply connected closed Riemannian manifold $M$ and the dimension of rational homotopy $\dim_{\mathbb{Q}}...
5
votes
0answers
85 views
Proof of MacPherson's result about set-valued constructible sheaves and exit paths
I'm looking for a proof of a theorem that is attributed to MacPherson. Treumann (Section 1.1 in Exit paths and constructible stacks, 2009) states the theorem as:
Theorem 1.2 (MacPherson). Let $(X,S)...
5
votes
0answers
67 views
Products of representables are regular on a regular skeletal Reedy category?
The definition of a (pre)regular skeletal Reedy category in the sense of Cisinski generalizes the intuition behind the Eilenberg-Zilber factorization for simplicial sets, specifically the property ...
6
votes
1answer
199 views
Is there a relationship between norms/transfers in equivariant homotopy theory and norms in the Tate construction / ambidexterity?
In homotopy theory, the word "norm" is commonly used in two different ways (well, surely there are other ways, but these two have a particular familial resemblence).
Let $G$ be a finite group. A $G$-...
10
votes
2answers
545 views
What is the best reference for motives?
I want to learn about homotopy theory on number fields, and I heard that the theory of motives made it possible, so I want to know what is a good textbook for motive theory.
To be honest, I don’t ...
10
votes
1answer
418 views
Poincare duality spaces vs. manifolds via lifting maps, the obstruction theory and the role of simply connectedness
Suppose that we are given a topological space $X$: assume for simplicity that $X$ is compact we want to adress the following question:
Is it true that one can find a manifold $M$ which is homotopy ...
4
votes
0answers
74 views
Classification of combinatorial model categories presentable by simplicial presheaves on a Reedy category
Dan Dugger proved that every combinatorial model category can be obtained up to Quillen equivalence as the localization of a model structure on simplicial presheaves on a small category $C$.
Is there ...
7
votes
1answer
120 views
When does a map of spaces deloop a closed subgroup inclusion?
I believe Kan showed that any connected CW complex is the delooping of a topological group. I'm interested in the relative question:
Question: Let $Y \to X$ be a map of connected CW complexes. Under ...
5
votes
1answer
119 views
Model category of diagrams with the colimit detecting the weak equivalences
Let $I$ be a small category and $\mathcal{K}$ be a combinatorial model category.
Is it known a model category structure on the functor category
$\mathcal{K}^I$ such that a map of diagrams $D\to ...
4
votes
1answer
113 views
Trivial cohomology with free module coefficient
Let $G$ be a group and $M$ be a free $\mathbb{Z} G$-module. Then $H^2(G,M)=0$. Is this statement correct?
I know that if $M$ is injective module, then $H^n(G,M)=0$ for all $n\geq 1$. But I have no ...
7
votes
1answer
103 views
Homotopy pushout independent of factorization and symmetric in cofibration category
$\require{AMScd}$In Algebraic homotopy, Baues defines the notion of homotopy pushout in a cofibration category in the following way: a commutative diagram
\begin{CD}
A @>k>> C \\
@AfAA @AAhA\...
4
votes
0answers
270 views
Nonabelian Poincare duality
Nonabelian Poincare duality is introduced by Jacob Lurie in "Higher Algebra", Section 5.5.6.
It seems, that my question is closely related to this definition.
Question: what can one say about the ...
8
votes
0answers
315 views
Floer cohomology from mapping spaces of $\infty$ categories
There's a meta-observation (of Urs Schreiber, who attributes it to Ken Brown and Lurie) that 'cohomology theories come from mapping spaces of $(\infty,1)$ categories'. This is described in detail at ...
14
votes
1answer
610 views
Homotopy pullback of a homotopy pushout is a homotopy pushout
Let's assume that we have a cube of spaces such that everything commutes up to homotopy.
The following holds:
- The right square is a homotopy pushout and
- all the squares in the middle are ...
18
votes
0answers
447 views
Eckmann-Hilton argument / Grothendieck
In terse terms, the “Eckmann-Hilton argument” says that a monoid in the category of monoids is commutative. It is essentially used, for example, to prove that homotopy groups of topological groups are ...
5
votes
0answers
165 views
Asphericity of hypersurface complement in ${\mathbb C}^n$
How does one check that the following space is aspherical?
$X_n=\{(x_1,x_2,\ldots , x_n)\in {(\mathbb C^*)}^n\ |\ x_i\neq x_j\ and\ x_ix_j\neq 1\ for\ i\neq j\}$.
One way I can think of is to give ...
8
votes
1answer
119 views
Cofibrations in a category of fibrant objects
There is an obvious (?) notion of cofibration in a category of fibrant objects, namely a morphism which satisfies the left lifting property with respect to all trivial fibrations. I don't seem to be ...
13
votes
0answers
206 views
Localizing $\mathrm{CombModCat}$ at the Quillen equivalences
Let $\mathrm{CombModCat}$ be the category of combinatorial model categories with left Quillen functors between them. By Dugger's theorem and the appendix of Lurie's "Higher Topos Theory" it ought to ...
12
votes
1answer
298 views
Which motivic spectra are dualizable?
Let $S$ be a scheme, and $SH(S)$ the stable motivic category over $S$. Which objects of $SH(S)$ are dualizable with respect to the smash product?
All I can find on this question is an old abstract of ...
2
votes
1answer
374 views
A projective (or free) $\mathbb{Z}\pi_1$-module
Suppose that $Z$ is a finite wedge of spheres containing circles and there exist maps $f:Y\to Z$ and $g:Z\to Y$ so that $g\circ f\simeq 1_Y$. Assume that there exists a map $h:X\to Y$ which induces ...
3
votes
2answers
227 views
Pushout of spaces
Suppose that we have a map between two pushout diagrams of topological spaces
$$ [A_{1}\leftarrow A_{0}\rightarrow A_{2} ] \rightarrow [B_{1}\leftarrow B_{0}\rightarrow B_{2} ]$$
such that for any ...
1
vote
1answer
222 views
pushout and homotopy
Suppose that we have tree spaces $A, B, C$ (let say CW-complexes). Are given two pairs of maps:
$$f_{0},g_{0}: A\rightarrow B $$
$$f_{1}, g_{1}: A\rightarrow C $$
such that $f_{0},g_{0}$ are ...
6
votes
2answers
260 views
Are cofibrations accessible?
The category of fibrations in a combinatorial model category is accessible, accessibly embedded in the arrow category. How about the cofibrations?
More generally, let $C$ be a locally presentable ...
3
votes
0answers
140 views
Comparing two simplicial resolution in a model category
Suppose that is a nice model category $\mathbf{M}$ (cofibrantly generated,...).
Suppose we have a two diagrams
$$F,G: \Delta^{op}\rightarrow \mathbf{M} $$
and a natural transformation $\nu: F\...
9
votes
0answers
154 views
Is there an “injective version” of the Bergner model structure?
The Bergner model structure on $sCat$ (simplicially enriched categories) has a "projective" flavor: fibrations are "levelwise" while cofibrations satisfy stringent relative "freeness" conditions.
...
3
votes
0answers
75 views
What is the structure required to construct this homotopy of maps between mapping cones?
Let $X,Y$ be topological spaces, let $f$ be a continuous map from $X$ to $Y$ and let $g$ be a continuous map from $Y$ to $X$. Write $C_f$ for the mapping cone of $f$; i.e., $\{*\} + X\times I + Y$, ...
4
votes
0answers
65 views
Contracting the rational bar cocomplex for a finite group G
Let $G$ be a finite group and define $B^p(G,\mathbb{Q}) = {\rm Functions}(G^p,\mathbb{Q})$. These $\mathbb{Q}$-vector spaces assemble into a cochain complex with differential
$$d \sigma(g_0,\dots,g_p) ...
11
votes
1answer
331 views
Formality of the 2nd ordered configuration space of a closed Riemann surface
If $X$ is a smooth manifold, we define its kth ordered configuration space as $$F_kX:=\{(x_1, \ldots,x_k) \; | \; x_i \neq x_j \,\, \mathrm{if} \, \, i \neq j\},$$
in other words, $F_kX = X^k - \Delta,...
21
votes
1answer
623 views
Is algebraic $K$-theory a motivic spectrum?
I've received conflicting messages on this point -- on the one hand, I've been told that "forming a natural home for algebraic $K$-theory" was one motivation for the development of motivic homotopy ...
5
votes
0answers
115 views
Applications of spectral Artin representability?
The spectral Artin representability theorem says that a functor $X:CAlg^{cn}\rightarrow S$ from the $\infty$-category of connective $E_{\infty}$-rings to the $\infty$-category of small topological ...
4
votes
1answer
138 views
A question regarding generalized cohomology and spectra : proof of $E^{\ast}(S)\otimes\mathbb{R} = H^{\ast}(S;\pi_{\ast}E\otimes \mathbb{R})$
I asked a question on m.se about generalised cohomology and spectra. Not having received any specific answer I attempted to draw more attention by offering a bounty. But I still could not get any help....
2
votes
0answers
142 views
On the paradox that $n$-coskeletal simplicial sets model all homotopy types
Please help me resolve the following paradox:
False claim: Let $X$ be an $n$-coskeletal, $n$-connected simplicial set. Then $X$ is weakly contractible.
Actually, I suppose the claim is ...
7
votes
1answer
209 views
What is the relationship between $O_G$-parameterized $\infty$-categories and $\infty$-categories enriched in $Top_G$?
Let $G$ be a finite group. Barwick et al define a $G-\infty$-category to be a fibration over the orbit category $O_G$ of transitive $G$-sets. But in the non-$\infty$-land, the natural guess at where I ...
