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 ...
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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 ...
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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\...
- 613
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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$?
- 21
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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....
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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 $...
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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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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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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{...
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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 ...
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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 ...
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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 ...
- 1,873
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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 $\operatorname{Ho}(sSet).$ ...
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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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$\pi_k(\mathbb{S}^n\vee\mathbb{S}^n)$
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 $...
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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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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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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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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 ...
- 1,143
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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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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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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 ...
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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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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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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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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 ...
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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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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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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,...
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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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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. ...
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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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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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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/...
- 1,143
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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 ...
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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 ...
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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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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 ...
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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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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-...
- 1,143
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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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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}.$$
At the heart of homotopy theory ...
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Higher Topos Theory- what's the moral?
I've often seen Lurie's Higher Topos Theory praised as the next "great" mathematical book. As someone who isn't particularly up-to-date on the state of modern homotopy theory, the book seems ...
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Monoidal Dold–Kan correspondence for non-connected CDGA
Why can't the monoidal Dold–Kan correspondence be extended to non-connected CDGAs over a field of characteristic 0?
I understand that there is a technical problem with the original proof due to ...
- 613
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On morphism spaces of $(\infty,2)$-categories failing to be $\infty$-categories
As noted in Tag 01WZ of Kerodon, the morphism spaces of an $(\infty,2)$-category $C$ can fail to be $\infty$-categories, in contrast to the pinched morphism spaces of $C$.
This feels very ...
- 1,143
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208
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Homotopy theory of cospaces (or $\infty$-cogroupoids)
Is there a good homotopy theory for cospaces, where a cospace (or $\infty$-cogroupoid) would be a cosimplicial set satisfying some appropriate dual version of the Kan condition?
One point I'm curious ...
- 1,143
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What is the Goodwillie calculus interpretation of Quillen's rational homotopy theory?
$\newcommand\Spaces{\mathit{Spaces}}\newcommand\sLie{\mathit{sLie}}\DeclareMathOperator\id{id}$Let $X$ be a space. Then $\pi_\ast(X)$ is a shifted Lie algebra under the Whitehead bracket $[-,-]$. ...
- 53.6k
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Maps in the slice category vs. maps in the arrow category
Let $f:x\to z$ and $g:y\to z$ be morphisms in an $\infty$-category $\mathcal C$. It seems that the square
$$\require{AMScd}
\begin{CD}
\operatorname{Map}_{\mathcal C_{/z}}(f,g) @>>> \...
- 1,137
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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 ...
- 53.6k