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 ...
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2
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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 ...
7
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2
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462
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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 ...
7
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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(...
7
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1
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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 ...
7
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2
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458
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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 ...
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511
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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\}...
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1
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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 ...
7
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3
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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 ...
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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 ...
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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$^4$...
7
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2
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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 ...
7
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1
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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 ...
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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.
...
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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 ...
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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
$$\rho:SU(2)\...
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Why does $\iota_4^2 \in H^8(K(\mathbb Z/2,4);\mathbb Z/2)$ not come from $H^8(K(\mathbb Z/2,4);\mathbb Z)$?
In Hatcher's Chapter 5 (https://pi.math.cornell.edu/~hatcher/AT/ATch5.pdf) on page 574 (page 57 in the pdf), he states that $\iota_4^2 \in H^8(K(\mathbb Z/2,4);\mathbb Z/2)$ is not in the image of $H^...
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3
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Classification of sections of free loop fibration over the two-sphere
For any space $X$ there is a fibration
$$
\Omega X\to LX\stackrel{ev}{\to} X
$$
where $LX=Map(S^1,X)$ is the free loop space, $\Omega X = Map_*(S^1,X)$ is the based loop space, and $ev:LX\to X$ is the ...
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Etale and Algebraic K-theory with rational coefficients
We know that the Quillen-Lichtenbaum conjecture, now proved by Rost, Voevodsky, and Weibel, says that for smooth finite type $k$-schemes $X$, etale and algebraic $K$-theory with finite coefficient $\...
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Can a weak fibration category be non saturated?
A weak fibration category is a category $\mathcal{C}$ equipped with two subcategories
$$\mathcal{F}, \mathcal{W} \subseteq \mathcal{C}$$
containing all the isomorphisms, such that the following ...
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1
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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 $\...
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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 ...
7
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1
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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 ...
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1
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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 ...
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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 $...
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2
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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-...
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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]$, ...
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2
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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 ...
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3
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}}{=...
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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 $\...
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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? ...
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1
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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 ...
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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 ...
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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 ...
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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 $...
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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 $Z$, ...
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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 ...
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1
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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 ...
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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 ...
7
votes
1
answer
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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 $HA$,...
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1
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460
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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 ...
7
votes
1
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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, ...
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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 ...
7
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1
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724
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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 ...