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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How to learn homotopy theory

I studied some basic algebraic topology (homotopy/homology/cohomology groups). When reading about the Dold-Thom theorem, the fancier and more recent sources sooner or later all started to use homotopy ...
5 votes
1 answer
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Rational G-spectrum and geometric fixed points

For a finite group $G$, how is a rational $G$-spectrum $X$ detected by the geometric fixed point functor $\phi^H$ where we consider the conjugacy class of $H\leq G$? I tried finding a reference for ...
1 vote
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Explicit proof of Quillen's connectivity theorem

Definition Let $A$ be a commutative ring. An ideal $I \triangleleft A$ is called quasiregular if $I/I^2$ is flat over $A/I$ and there is a canonical isomorphism of algebras $$ \Lambda_A I/I^2\...
2 votes
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Snaith Theorem and connective K-theory

The Snaith Theorem tells us $KU \simeq \mathbb{S}[\mathbb{C}P^{\infty}][\beta^{-1}]$. Is it possible to give a related description of connective k-theory $ku$?
3 votes
2 answers
187 views

Uniformly continuous homotopy equivalence

Suppose $M$ and $N$ are complete metric spaces and $f, g: M \to N$ are uniformly continuous maps between them with common modulus of continuity $m$. Further suppose $f$ and $g$ are homotopy equivalent....
5 votes
2 answers
273 views

Construct a 'nice' trivializing cover of universal principal $G$-bundle $EG \to BG$

Let G be a discrete or say for sake of simplicity a finite group. In Hatcher's book Algebraic Topology on p 89 the construction of universal bundle $EG$ carries structure of a $\Delta$-complex whose $...
2 votes
0 answers
181 views

About infinite loop space and $\Omega$ spectrum

Let $A$ is an topological abelian monoid. Also $\pi_0(A)$ is a group and $A$ has $CW$ structure. $BA$ is a classifying space of the topological abelian monoid. My purpose is to construct an infinite ...
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4 votes
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116 views

The homotopy type of the space of symplectic structures

While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures ...
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1 vote
1 answer
148 views

fundamental group of $X/\mathbb{R}^n$

Suppose with have a topological manifold $X$ and a group $G$, is there a way to compute the fundamental group of $X/G$ in function of $\pi(X)$ and $\pi(G)$? are there any settings on X that can ...
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Name for homotopy totalization of Goodwillie tower (in embedding calculus)

Let $M,N$ be a manifold and consider the presheaf of spaces $\textrm{Emb}(-, N)$ on the open sets of $M$. Classical embedding calculus produces a goodwillie tower $$ \ldots \rightarrow T_{k+1} \textrm{...
2 votes
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Characterization of growth in terms of coarse algebraic topology

$$ \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mbb}[1]{\mathbb{#1}} \newcommand{\opn}[1]{\operatorname{#1}} \DeclareMathOperator\cap{cap} \def\sse{\subseteq} $$ Coarse spaces Let $X$ be a coarse ...
4 votes
1 answer
308 views

Is the concept of an $H$ object still interesting, when we have the $\infty$-version of it?

Recently I got acquainted with $\infty$-algebraic theories. I expect $\infty$-algebraic objects (of ordinary Lawvere theories) in $\infty\text{-}\mathrm{Groupoid}$ to behave much better than algebraic ...
9 votes
1 answer
232 views

Does every complex orientable $E_\infty$-ring admit an $E_\infty$ complex orientation?

A ring spectrum $E$ is complex oriented if it is equipped with a ring map $MU\rightarrow E$. It is complex orientable if such a ring map exists. An $E_\infty$-ring $E$ is $E_\infty$-complex oriented ...
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8 votes
1 answer
187 views

When a model category with prescribed homotopy category exists?

My question is probably insanely hard or very well-studied, but I could not find an answer, so I will ask it here. Assume that we have a suitably complete closed module over $\operatorname{Ho}(sSet).$ ...
11 votes
1 answer
393 views

A topological tree is weakly contractible

Let us call a nonempty topological space a topological tree if it is Hausdorff and for two distinct points there is a continuous injective path connecting the points, which is unique up to ...
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12 votes
1 answer
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$\pi_k(\mathbb{S}^n\vee\mathbb{S}^n)$

According to Hilton-Milnor theorem for $n\geq 2$ $$ \pi_k(\mathbb{S}^n\vee\mathbb{S}^n)= \pi_k(\mathbb{S}^n)\oplus \pi_k(\mathbb{S}^n)\oplus \bigoplus_{i=1}^\infty \pi_k(\mathbb{S}^{m_i}), $$ where $...
4 votes
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332 views

Are Frobenius modules related to Frobenius algebras?

Frobenius modules appear in the Riemann Hilbert correspondence. Frobenius algebras appear in TQFT. Is there a way to pass from one to the other?
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3 votes
1 answer
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Can the Picard-graded homotopy of a nonzero object be nilpotent?

Let $\mathcal C$ be a symmetric monoidal stable category such that the thick subcategory generated by the unit is all of $\mathcal C$ -- in particular, every object is dualizable (I'm particularly ...
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1 vote
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58 views

bott element in periodic cyclic homology

I am reading a paper by Thomason "Algebraic K-theory and étale cohomology", which deals with algebraic K theory localized by inverting the bott element $\beta \in K_2(\overline{k})^{\wedge}...
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5 votes
1 answer
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Inexistence of a Kan–Quillen model structure on globular sets

(This is in a sense a follow-up to my earlier question on a geometric definition of globular $\infty$-groupoids) We know by Scholie 8.4.14 of Cisinski's thesis that the globe category $\mathbb{G}$ is ...
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5 votes
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186 views

Cyclic homology can be recovered from topological cyclic homology?

Let $R$ be a commutative ring, $S$ a $R$-algebra of finite type. By an equivalence of ring spectra $$ \operatorname{HH}(S/R) \simeq \operatorname{THH}(S)\otimes_{\operatorname{THH}(R)} HR, $$ ...
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4 votes
1 answer
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If $A_\ast$ has a Künneth theorem, then is $A$ a module over Morava $K$-theory?

$\newcommand\Spt{\mathit{Spt}}\newcommand\GrAb{\mathit{GrAb}}$Let $A$ be a ring spectrum. Suppose that $A$ has a Künneth theorem — i.e. the homology theory $A_\ast : \Spt \to \GrAb$ is a strong ...
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4 votes
1 answer
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Decomposing a $\mathcal{M}$-valued presheaf into a homotopy colimit of representables

The context for this question comes from this arxiv preprint. Specifically, a remark in the final proof of the paper. To make the question more self-contained, I'll phrase this question in a slightly ...
1 vote
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169 views

The key step in Serre's method on higher homotopy groups

Let $n \geq 2$ and $X$ be a $(n-1)$-connected simplicial complex. This means that all of the lower homotopy groups $\pi_{k}(X) = 0$ for $k \leq n-1$. My goal is to compute the higher homotopy groups ...
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3 votes
1 answer
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Is $\Sigma^\infty_+ O(n)^\vee$, the Spanier-Whitehead dual of the orthogonal group, an $A_\infty$-ring spectrum?

Recently, Ching and Salvatore have proven that the $E_n$ operad is Koszul self dual. While thinking about the analogous question for the framed $E_n$ operad, I realized there is an obvious first ...
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5 votes
1 answer
217 views

If $\pi_\ast A$ is graded-commutative, then is $A_\ast$ a lax monoidal functor?

Let $A$ be a homotopy ring spectrum. Then the homology theory $A_\ast : Spectra \to GrAb$ lifts to a homology theory valued in $GrMod(\pi_\ast A)$. If $A$ is homotopy commutative, then this functor $...
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7 votes
0 answers
137 views

Relative version of Browder's theorem on H-spaces

A theorem of W. Browder [1] from 1961 says that an H-space $X$ with finitely generated integral homology (meaning that $H_*(X)$ is finitely generated) has $\pi_2(X) = 0$. This generalizes Cartan's ...
5 votes
0 answers
162 views

Flatness of objects in a prestable $\infty$-category

I wonder what is the correct concept of flatness of objects in a prestable $\infty$-category with appropriate conditions? The typical example is the following. Let $R$ be a connective $\mathbb E_1$-...
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2 votes
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282 views

Invariants of objects in $\operatorname{Ch}(\mathrm{Ab})$ up to chain homotopy

$\newcommand\Ab{\mathrm{Ab}}\newcommand\ab{\mathrm{ab}}\DeclareMathOperator\Ch{Ch}\DeclareMathOperator\Kom{Kom}\newcommand\ho{\mathrm{ho}}$Let $\Ab$ be the category of finitely generated abelian ...
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4 votes
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Is it possible to define the Serre mod $\mathscr{C}$ model category structure on $r$-reduced simplicial sets as Cisinski model structure?

I think it is quite straightforward to show that a $\bmod\mathscr{C}$ model structure on $r$-reduced simplicial sets is a Bousfield localization of the transferred Kan-Quillen model structure. However,...
1 vote
0 answers
112 views

Khovanov $A_\infty$ algebra

Let $L$ be a link in $\mathbb{R}^3$, with $D, D'$ be diagrams in $\mathbb{R}^2$ representing $L$. Khovanov constructed two graded chain complexes $$C_{D} = (Ch_{D}, d_{D}) \quad C_{D'}=(Ch_{D'}, d_{D'}...
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9 votes
1 answer
414 views

The center of $\mathbf{hTop}$

What is the center of the homotopy category $\mathbf{hTop}$? I strongly believe that it is trivial, but it is hard to prove since $\mathbf{hTop}$ is not concretizable and hence has no small separator. ...
4 votes
0 answers
86 views

For which operads $O$ does $\operatorname{coAlg}_O(C) = C$ whenever $C$ is cartesian monoidal?

Let $O$ be an operad, and let $D$ be a symmetric monoidal category. Then there is a forgetful functor $\operatorname{Alg}_O(D) \to D$. This functor is an equivalence in either of the following cases: ...
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3 votes
0 answers
141 views

Pushout homotopy squares in motivic homotopy theory

I am using Vladimir Voevodsky's notes on motivic homotopy theory and I am having trouble understanding the proofs of Corollary 2.20 and Lemma 2.21. Both of these deal with homotopy pushout squares and ...
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5 votes
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A theory of higher limits of (1-)functors, after higher hochschild homology

$\newcommand{\Trans}{\mathrm{Trans}}\newcommand{\H}{\mathrm{H}}\newcommand{\HH}{\mathrm{HH}}\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\Mod}{\mathrm{Mod}}$Recently I noticed that one may regard the co/...
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6 votes
1 answer
593 views

Domain of left adjoint from condensed sets to anima

$\DeclareMathOperator\Hom{Hom}$Let $X$ be a condensed set in the sense of Clausen-Scholze. If there is a universal anima $Y$ (or $\infty$ groupoid, or homotopy type) together with a map of condensed ...
13 votes
1 answer
414 views

Applications of equivariant homotopy theory to representation theory

Equivariant homotopy theory focuses on spaces together with some group action on them. Jeroen van der Meer and Richard Wong have a paper where they use equivariant methods to compute the Picard group ...
4 votes
1 answer
202 views

When are filtered colimits of (trivial) cofibrations still (trivial) cofibrations?

Let $\mathcal{M}$ be a locally finitely presentable model category, cofibrantly generated by two sets $\mathcal{I}$ and $\mathcal{J}$ of cofibrations and trivial cofibrations with presentable domain ...
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3 votes
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subcategory of "nice" maps of topological spaces where each closed inclusion is a cofibration?

Is there a subcategory of topological spaces such that each closed inclusion is necessarily a cofibration, and which is good for homotopy theory ? In general, what are sufficient conditions for a ...
3 votes
1 answer
228 views

Homology of braid groups and loop spaces

How do Segal's theorems from (Configuration-spaces and iterated loop-spaces. Invent. Math.21:213--221) imply that there is an isomorphism $H_*(B_\infty,\mathbb{Z})\cong H_*(\Omega^2S^3,\mathbb{Z})$, ...
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6 votes
1 answer
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Do the various notions of morphism spaces of simplicial sets agree on the underived level?

$\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}$There are (at least) seven kinds of morphism spaces for a simplicial set $X$: The left-pinched morphism space $\Hom^L_X(x,y)$, The right-...
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6 votes
0 answers
175 views

Functorial identification of the mapping spaces of the arrow category of an $\infty$-category

Let $\mathcal{C}$ be a small $\infty$-category. Using the straightening-unstraightening construction, we can define a hom-functor of $\mathcal{C}$ as a functor $\mathcal{C}(-,-):\mathcal{C}^{\mathrm{...
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1 vote
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Is the equivalence $\mathrm{CRing}^{\mathrm{op}}\simeq \mathrm{AffSch}$ related to the homotopy hypothesis?

At the heart of algebraic geometry lies the op-equivalence between commutative rings and affine schemes, i.e., $$\mathrm{CRing}^{\mathrm{op}}\simeq \mathrm{Aff\,Sch}.$$ At the heart of homotopy theory ...
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25 votes
3 answers
4k views

Higher Topos Theory- what's the moral?

I've often seen Lurie's Higher Topos Theory praised as the next "great" mathematical book. As someone who isn't particularly up-to-date on the state of modern homotopy theory, the book seems ...
4 votes
1 answer
173 views

Monoidal Dold–Kan correspondence for non-connected CDGA

Why can't the monoidal Dold–Kan correspondence be extended to non-connected CDGAs over a field of characteristic 0? I understand that there is a technical problem with the original proof due to ...
6 votes
1 answer
213 views

On morphism spaces of $(\infty,2)$-categories failing to be $\infty$-categories

As noted in Tag 01WZ of Kerodon, the morphism spaces of an $(\infty,2)$-category $C$ can fail to be $\infty$-categories, in contrast to the pinched morphism spaces of $C$. This feels very ...
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4 votes
0 answers
208 views

Homotopy theory of cospaces (or $\infty$-cogroupoids)

Is there a good homotopy theory for cospaces, where a cospace (or $\infty$-cogroupoid) would be a cosimplicial set satisfying some appropriate dual version of the Kan condition? One point I'm curious ...
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9 votes
0 answers
231 views

What is the Goodwillie calculus interpretation of Quillen's rational homotopy theory?

$\newcommand\Spaces{\mathit{Spaces}}\newcommand\sLie{\mathit{sLie}}\DeclareMathOperator\id{id}$Let $X$ be a space. Then $\pi_\ast(X)$ is a shifted Lie algebra under the Whitehead bracket $[-,-]$. ...
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4 votes
1 answer
165 views

Maps in the slice category vs. maps in the arrow category

Let $f:x\to z$ and $g:y\to z$ be morphisms in an $\infty$-category $\mathcal C$. It seems that the square $$\require{AMScd} \begin{CD} \operatorname{Map}_{\mathcal C_{/z}}(f,g) @>>> \...
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3 votes
0 answers
260 views

Higher-order HKR theorems?

Recall that Hochschild-Kostant-Rosenberg -type theorems identify certain smoothness conditions guaranteeing an isomorphism between the cotangent complex and (a shift of) the Hochschild homology of an ...
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