Questions tagged [homotopy-theory]

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 and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.

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An explicit description of Lawvere's segment in the category of simplicial sets

In any presheaf topos, there exists an object called Lawvere's segment, which can be described as the presheaf $L:A^{op}\to Set$ such that for each object $a\in A$, $L(a)=\{x\hookrightarrow\ h_a: x\in ...
Harry Gindi's user avatar
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Rational Group Cohomology

This is a general question about group cohomology. I'm interested in the case when the coefficients are the rational numbers and hence I suppose when my groups are infinite. The question splits into ...
James Griffin's user avatar
7 votes
2 answers
462 views

Brown representability in slice category

Brown's representability theorem gives us a very nice set of conditions to check that a (contravariant) functor $Hot^{op}\rightarrow Set$ is representable. Choose an object $X$ in $Hot$. Then it is ...
curious math guy's user avatar
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2 answers
3k views

the hopf invariant of the hopf construction

I'm having some trouble with a problem about the Hopf construction, in the exercises for Ch. 4 of Mosher & Tangora. Given a map $g : S^{n-1} \times S^{n-1} \rightarrow S^{n-1}$, we get a map $h(...
Aaron Mazel-Gee's user avatar
7 votes
1 answer
584 views

Motivation for the definition of complex orientable cohomology theory

PRELIMINARY DEFINITIONS: Let $E^*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have: $$ \tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt) $$ So there is a special ...
Tommaso Rossi's user avatar
7 votes
2 answers
458 views

Trees in chain complexes

$\DeclareMathOperator{\Ch}{\mathit{Ch}}$Let $\Ch_\mathbb{Q}$ denote the model category of chain complexes over rational numbers. Let $T_\ast$ be a tree in $\Ch_{\mathbb{Q}}$ with $n$ vertices. How to ...
Surojit Ghosh's user avatar
7 votes
2 answers
511 views

Chromatic t-structures?

Questions: Fix a prime $p$ and $n \in \mathbb N_{\geq 1}$. Does the category $Sp_{K(n)}$ of $K(n)$-local spectra admit a nontrivial $t$-structure? By "nontrivial", I simply mean that $\{0\}...
Tim Campion's user avatar
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Is the Thomason model structure the optimal realization of Grothendieck's vision?

In Pursuing Stacks, Grothendieck uses the category $Cat$ of small categories to model spaces. A recurring theme is the question of whether there is a Quillen model structure supporting this homotopy ...
Tim Campion's user avatar
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7 votes
3 answers
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Is $S^1\vee S^1$ an Eilenberg-Mac Lane Space to a Homotopy Purist?

I believe that $S^1\vee S^1$ is the Eilenberg-Mac Lane space $K(\mathbb{Z}\ast\mathbb{Z},1)$. One can prove this by constructing its universal cover and observing that it is contractible. My question ...
Tyrone's user avatar
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Homologically distinct infinite loop structures on a space

Let $X$ be a connected pointed topological space equipped with two different actions of $E_\infty$-operad. Each action provides a collection of deloopings $X_i$, where $X_0 = X$ and $\Omega X_i$ is ...
Dmitry Pirozhkov's user avatar
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1 answer
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A problem on infinite dimensional metric space

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: $X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$...
Sebastien Palcoux's user avatar
7 votes
2 answers
516 views

Chromatic convergence of E(n)-localized homotopy categories

Given the Chromatic Convergence Theorem, can we state this globally as some convergence in the category of categories? That is, we have the subcategories of finite spectra in each stable homotopy ...
Jonathan Beardsley's user avatar
7 votes
1 answer
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Model categories as a tool to resolve size issues for localizing categories

I have a rather basic question about one motivation for introducing model categories in David White's notes, as a possible way to overcome troubles with size issues appearing when localizing a ...
user267839's user avatar
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Indexing categories of derivators

It is not clear to me the role of the domain and target in the definition of prederivators. For instance, the classical references put the domain as $\mathit{Dia}$, others as $\mathit{Cat}$ itself. ...
user234212323's user avatar
7 votes
1 answer
411 views

Proper homotopy

Let $F: X \times [0, 1] \to Y$ be a homotopy such that for any $t \in [0,1]$ the map $F( \cdot, t) : X \to Y$ is proper. Is it true in general that $F$ is proper? I am interested in particular in ...
Onil90's user avatar
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1 answer
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Isomorphism from $\mathbb{Z}$ to third homotopy group of compact simple Lie group

Let $G$ be a compact connected simple Lie group. It is known that its third homotopy group $\pi_3(G)$ is isomorphic to $\mathbb{Z}$. More precisely, there is a Lie group homomorphism $$\rho:SU(2)\...
user104853's user avatar
7 votes
1 answer
190 views

Why does $\iota_4^2 \in H^8(K(\mathbb Z/2,4);\mathbb Z/2)$ not come from $H^8(K(\mathbb Z/2,4);\mathbb Z)$?

In Hatcher's Chapter 5 (https://pi.math.cornell.edu/~hatcher/AT/ATch5.pdf) on page 574 (page 57 in the pdf), he states that $\iota_4^2 \in H^8(K(\mathbb Z/2,4);\mathbb Z/2)$ is not in the image of $H^...
Dolly Wu's user avatar
7 votes
3 answers
367 views

Classification of sections of free loop fibration over the two-sphere

For any space $X$ there is a fibration $$ \Omega X\to LX\stackrel{ev}{\to} X $$ where $LX=Map(S^1,X)$ is the free loop space, $\Omega X = Map_*(S^1,X)$ is the based loop space, and $ev:LX\to X$ is the ...
Mark Grant's user avatar
7 votes
1 answer
785 views

Etale and Algebraic K-theory with rational coefficients

We know that the Quillen-Lichtenbaum conjecture, now proved by Rost, Voevodsky, and Weibel, says that for smooth finite type $k$-schemes $X$, etale and algebraic $K$-theory with finite coefficient $\...
user97282's user avatar
7 votes
1 answer
182 views

Can a weak fibration category be non saturated?

A weak fibration category is a category $\mathcal{C}$ equipped with two subcategories $$\mathcal{F}, \mathcal{W} \subseteq \mathcal{C}$$ containing all the isomorphisms, such that the following ...
Ilan Barnea's user avatar
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Is it known whether this space is a suspension space?

For each prime $p\geq 3$ let $\alpha_p:S^{2p}\to S^3$ denote a representative of $\pi_{2p}S^3$ of order $p$. Berstein and Hilton showed that for each $p$ the homotopy cofiber $C_{\alpha_p}$ of $\...
James Schwass's user avatar
7 votes
1 answer
698 views

Fiber vs homotopy fiber in model categories: simple question

I have a concrete problem with the homotopy fiber and I am getting lost with the literature. I state my question and, to avoid confusions, I state downwards the definitions I am using. Let $C$ be a ...
Tintin's user avatar
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7 votes
1 answer
179 views

If a loopspace admits space-level power operations, is is a higher loopspace?

Let $G$ be a group. Suppose that the map $G \to G$, $g \mapsto g^r$ is a group homomorphism for every $r \in \mathbb N$. Then $G$ is abelian. Is this also true homotopy-theoretically? (In the ...
Tim Campion's user avatar
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7 votes
1 answer
438 views

Reference request for K-Theory linearization

I posted this question on math.se here:https://math.stackexchange.com/q/2996787/482732, but I think it may be more appropriate here, sorry if I am wrong about that. In Waldhausen's paper Algebraic K ...
Noah Riggenbach's user avatar
7 votes
2 answers
292 views

Homotopy domination of a wedge of two polyhedra

The topological space $A$ is called homotopy dominated by the space $X$ if there are maps $f:A\longrightarrow X$ and $g:X\longrightarrow A$ so that $g\circ f\simeq id_A$. Question: Suppose that $...
M.Ramana's user avatar
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7 votes
2 answers
514 views

Explicit generating acyclic cofibrations and right properness of a model category

Let $\mathcal{C}$ be a cofibrantly-generated model category. My impression is that the following two conditions are highly correlated: $\mathcal{C}$ is right proper. There is an explicitly-...
Tim Campion's user avatar
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7 votes
1 answer
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classifying maps of Whitney sums of vector bundles

For an $n$-dimensional vector bundle $\xi$ with structure group $G\leq O(n)$ over a $CW$-complex $B$, we have a classifying map up to homotopy $$ f(\xi): B\longrightarrow BG, $$ $f(\xi)\in [B;BG]$, ...
Quan's user avatar
  • 519
7 votes
2 answers
397 views

Properness of the category of modules over a spectrum (that represents algebraic cobordism or motivic cohomology)

The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper? Since weak ...
Mikhail Bondarko's user avatar
7 votes
3 answers
884 views

A fibrant-objects structure on Top

(Sorry for the crossposting, but I'm really interested in this question). One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...
fosco's user avatar
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7 votes
1 answer
395 views

What is the homotopy type of a free simplicial ring?

Is there a good description of the homotopy type of a free simplicial ring (or simplicial $R$-algebra) on a given simplicial set, in terms of the homotopy type of that simplicial set? (This is mostly ...
Akhil Mathew's user avatar
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7 votes
2 answers
1k views

homotopy pushout of spaces homotopic to finite CW complexes

Does anyone know a reference for the fact that a homotopy pushout (double mapping cylinder) of spaces which are homotopy equivalent to finite CW complexes is also homotopy equivalent to a finite CW ...
Jim McClure's user avatar
7 votes
1 answer
751 views

A Model Structure on Symmetric Monoidal Categories

The recent article found here revisits Thomason's proof that symmetric monoidal categories model all connective spectra, but stops short of showing that there is a full closed model structure on this ...
Eric Finster's user avatar
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7 votes
1 answer
552 views

Homotopy orbit spaces of representation spheres

Let $G$ be a finite group and $V$ be finite-dimensional real representation of $G$. Write $S^V$ for the one-point compactification of $V$, with induced $G$-action, viewed as a pointed space, and ...
Reid Barton's user avatar
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7 votes
1 answer
326 views

How do the various homotopy 2-categories compare?

There are various models of $\infty$-categories floating around, so there are as many models of the associated homotopy 1- and 2-categories. Because the relations between the former are worked out in ...
Jonas Linssen's user avatar
7 votes
2 answers
861 views

Is there a "spectral exterior algebra" construction in higher algebra?

Given a ring spectrum $R$ and an $R$-module $E$, we have the spectral symmetric algebra $\mathrm{Sym}_R(E)$ of $E$ over $R$, defined by $$ \begin{align*} \mathrm{Sym}_R(E) &\overset{\mathrm{def}}{=...
Emily's user avatar
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7 votes
1 answer
410 views

Is there a topological interpretation of a module over $\Omega_{PL}(X)$?

Associated to a DGCA (differential graded commutative algebra) $A$, we can associate to it the category of modules over $A$. Hence, for a space $X$ we can consider the category of modules over $\...
Connor Malin's user avatar
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7 votes
1 answer
744 views

Colimits of cofibrations and homotopy colimits

Say C is a left proper model structure. I have a diagram where all maps are cofibrations. Is its colimit a homotopy colimit? I know this is true for pushouts. Is it true for sequential colimits? ...
naive-theorist's user avatar
7 votes
1 answer
521 views

Computing naive algebraic singular homology

I'm not sure if this counts as research level, since it might just be an expression of my ignorance. But anyway. Let $\Delta^n_A = Spec(A[X_0, \dots, X_n]/\sum X_i - 1)$ denote the standard algebraic ...
Tom Bachmann's user avatar
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7 votes
1 answer
436 views

How equivalent are the theories of reduced and groupal $\infty$-groupoids?

I hope that my question is sufficiently trivial that someone will be able to give me a pedantic answer, and not so trivial that no one takes the time to give an answer. My motivation for asking this ...
Theo Johnson-Freyd's user avatar
7 votes
3 answers
424 views

An obstruction theory for promoting homotopy equivalences that are equivariant maps to equivariant homotopy equivalences?

Say I have a map of $G$-spaces $f : X \to Y$ and I know it is a homotopy-equivalence in the plain sense that there exists a map (maybe not equivariant) $g : Y \to X$ such that the two composites are ...
Ryan Budney's user avatar
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7 votes
1 answer
175 views

Are morphisms in a stable $\infty$-category generated by split injections?

I've seen it stated in the $\infty$-categorical literature (without proof or reference) that every object in the $\infty$-category $\operatorname{Fun}(\Delta^1, \mathcal{C})$ of morphisms in a stable $...
Ishai Dan-Cohen's user avatar
7 votes
1 answer
535 views

Long exact sequences for parametrized cohomology

I'm reading Michael Shulman's articles on cohomology in HoTT here and here, as well as Floris van Doorn's thesis here. Given $E: Z \to \mathsf{Spectrum}$ a family of spectra over a homotopy type $Z$, ...
ಠ_ಠ's user avatar
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7 votes
1 answer
402 views

Does the small object argument need replacement?

Does one need the axiom of replacement in the small object argument and in the transfinite construction of free algebras? My motivation for the question is that I heard that the axiom of replacement ...
user333306's user avatar
7 votes
1 answer
220 views

Why are $G$-Extensions of fusion categories rigid, when constructed via monoidal 2-functors $G \to \underline{\underline{BrPic}}(\mathcal{D})$?

In arxiv:0909.3140 the $G$-extensions of a fusion category $\mathcal{D}$ are classified via monoidal 2-functors $G \to \underline{\underline{BrPic}}(\mathcal{D})$. A crucial part of the classification ...
Nicolas Cage's user avatar
7 votes
2 answers
302 views

Homotopicity of two certain sections of frame bundle of $GL(n,\mathbb{R})$

Edit: According to comment of Prof. GoodWillie we revise the question. Put $M=GL(n,\mathbb{R})$. We identify $M_n(\mathbb{R})$ with $\mathbb{R}^{n^2}$: The identification is based on the ...
Ali Taghavi's user avatar
7 votes
1 answer
1k views

Intuition - difference between Moore spectrum and Eilenberg-Mac Lane spectrum

I know very little about algebraic topology, and more about $k$-linear stable $\infty$-categories (i.e. homological algebra). Given an abelian group $A$, there is the Eilenberg-Mac Lane spectrum $HA$,...
Sasha's user avatar
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7 votes
1 answer
460 views

Commutativity up to homotopy implies strict commutativity, for lifting problems

Suppose we have a commutative diagram $\require{AMScd}$ \begin{CD} A @>>> X \\ @VVV & @VVV \\ W @>>> Y\\ \end{CD} where the map $A\rightarrow W$ is a cofibration and the ...
Diego95's user avatar
  • 511
7 votes
1 answer
259 views

Computation of a homotopy colimit of pro-spectra

Suppose $E\simeq\text{hocolim}_iE_i$ is a filtered homotopy colimit. Suppose $X=\{X_j\}_j$ is a pro-space (assume some finiteness conditions on the spaces or spectra if you have to...for example, ...
user108170's user avatar
7 votes
1 answer
262 views

Does every equivalence of operads in the category of small categories have a weak inverse?

Call a map of operads $\mathcal{O}\rightarrow \mathcal{U}$ in the category of small categories an equivalence, if each functor $\mathcal{O}(n)\rightarrow \mathcal{U}(n)$ is an equivalence of ...
simon's user avatar
  • 71
7 votes
1 answer
724 views

Thom isomorphism from the ABGHR perspective

In ABGHR Thom spectra are described in the following way: we start with a morphism of Kan complexes $X\to \mathbb{S}\text{-line}$, where $\mathbb{S}\text{-line}$ is an $\infty$-groupoid which is ...
Jonathan Beardsley's user avatar

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