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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Group Completion of a monoid (Braid groups)
Let $B_n$ be the braid group on $n$ strands, $B_{\infty}$ the direct limit of braid groups.
For a discrete group $G$, we let $BG$ to be the classifying space of $G$.
After reading this question, I was ...
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Shearing maps on domain of assembly map in algebraic $K$-theory
Let $H \to G$ be an inclusion of abelian groups, and let $R$ be a ${\Bbb Z}[H]$-algebra. Assume that the assembly map ${\Bbb S}[BG] \otimes_{\Bbb S} K(R \otimes_{{\Bbb Z}[H]} {\Bbb Z}[G]) \to K((R \...
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About the product of element of homotopy group [closed]
Why does the product coincide with the element naturally defined by oriented $S^2$ in https://arxiv.org/pdf/0801.3921.pdf page 29？
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Does there exist a Bousfield localization of the category of spectra which makes the sphere unbounded below?
Let $Sp$ be the category of spectra. Let $L : Sp \to Sp_L$ be the localization functor onto a reflective subcategory.
Question 1: Is it ever the case that $L(S^0)$ is not bounded below?
Question 2: ...
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Homotopy equivalence between certain loop spaces
I've been reading some papers carefully, with their proofs (Notations are given at the end).
The following comes from "Braids, mapping class groups and categorical delooping" by Song & ...
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Isbell duality for simplicial sets
$\newcommand{\IsbellSpec}{\mathsf{Spec}}\newcommand{\IsbellO}{\mathsf{O}}\newcommand{\Nat}{\mathrm{Nat}}$Isbell duality sets up an adjunction (see here for a short abstract summary)
$$\mathsf{O}\dashv\...
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What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
The difference between algebraic geometry and algebraic topology is that in AG, you usually hope that your spectral sequences degenerate immediately at the $E_2$ page. In AT, you often have to live ...
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Why is the universal $n$-gerbe universal? (HTT, 7.2.2.26)
Let $\mathcal{X}$ be an $\infty$-topos and let $A$ be an abelian group object of the category $\operatorname{Disc}(\mathcal{X})$ of discrete objects of $X$. Recall that a morphism $f:\widetilde{X}\to ...
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Does a ring spectrum with even homotopy and even cells always have a polynomial algebra of homotopy groups?
Let $R$ be a spectrum. Assume that $R$ is bounded-below. Then we can “even-ify” $R$: cone off some generating set of the first nonvanishing odd-dimensional homotopy group of $R$. Do this repeatedly. ...
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Left Kan extension and finite product preserving
Let ${\rm Ani(Ring)}$ be the $\infty$-category of animated commutative rings, ${\rm Ani(Ring)^{\leq 0}}$ be the category of discrete animated rings and ${\rm Ani}$ be the $\infty$-category of spaces . ...
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Homotopy 2-Category
I am reading Riehl's notes "Higher Category from scratch". I have come across the notion of a homotopy 2-category. I have not taken any course on Homotopy Theory before but I know that the ...
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Does the Atiyah-Hirzebruch spectral sequence for $E^\ast(X)$ collapse whenever $E$ is complex-oriented and $X$ has even cells?
Let $X$ be a (finite, say — or maybe of finite type) spectrum with even cells (in other words, $H_\ast(X;\mathbb Z)$ is free and concentrated in even degrees). Let $E^\ast$ be a complex-oriented ...
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Derivations in the Steenrod algebra
Let $\mathcal A^\ast$ be the (mod 2) Steenrod algebra.
Question 1:
Is there a classification of homogeneous elements $D \in \mathcal A^n$ such that $D^2 = 0$?
Question 2: Is there a classification of ...
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What is the center of Morava $K$-theory?
Let $E$ be an $E_1$ ring spectrum. Then I believe the center of $E$ is an $E_2$ ring spectrum over which $E$ is an $E_1$ algebra, given by the endomorphisms of $E$ as a bimodule over itself.
Question: ...
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Derived $\infty$-category of quasi-coherent sheaves on schemes
Let $X$ be a scheme. On the one hand, we have the derived $\infty$-category constructed from the abelian category of quasi-coherent sheaves on $X$. On the other hand, we can define the stable $\infty$-...
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homotopic to a constant map
Let $X$ and $Y$ be topological spaces and more precisely connected finite CW complexes.
Let $f\colon X \to Y$ be a continuous map such that there exist a second continuous map $F\colon X^3 \to Y$ and
$...
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Torus equivariant Morava K-theory
Let $X$ be a CW complex with a torus action $T$. Is there an established definition in equivariant stable homotopy theory of $T$-equivariant Morava K-theory, $K_p(n)^*_T(X)$? Any explicit references ...
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New investigations on Homotopical Algebraic Contexts
Homotopical algebraic context are models that allows Toën and Vezzosi to do derived geometry. It have been defined in their seminal paper Homotopical Algebraic Geometry II.
These are general abstract ...
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An open ended question: The dual of a covering map? Is this a real thing?
Reposted from this Reddit post as I didn't get good answers there:
So I've been reading about the Galois theory of covering maps and been staring at this equation for way too long:
$$\left| \pi_1(X,...
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Homotopical interpretation of Langlands correspondence
Recently I began learning about homotopy theory, I am very far from being familiar with all the basic notions and constructions, however I heard of the notion of topological modular forms. I also ...
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Homotopy theory and algebraic topology last 10 years. Is it a dying field? [closed]
I'm under the impression that algebraic topology is a dying field in mathematics. That was my impression but I think I'm wrong. As every person I do need some evidence that my impression is not ...
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Bar construction in commutative algebras is calculated by pushout
$\DeclareMathOperator\colim{colim}$
Also asked in MathStackexchange here
This is a statement in Lurie's Higher Algebra 5.2.2.4.
Proposition 3.2.4.7 in HA said that the monoidal structure on $\text{...
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$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$
This is a crosspost (with minor alterations).
For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category ...
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“Geometric” vs Homotopical completion
There are two notions of completions of slightly different nature, and I am wondering if there is a precise statement relating them.
The first one is the “homotopical” (or maybe it should be called ...
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Quasicompact quasiaffine classical schemes are nonconnectively-affine
In this answer to What is the relationship between connective and nonconnective derived algebraic geometry? I learned that any quasicompact open subscheme of an affine scheme is affine in the sense of ...
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$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]
If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of
$$
\left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
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Does there exist a closed 4-manifold whose $\pi_2$ contains torsion elements
Does there exist a closed 4-manifold $X$ such that $\pi_2(X)$ contains torsion elements?
And, if so, does there exist a closed 4-manifold $X$ such that $\pi_2(X)\neq 0$ but $\pi_2(X)\otimes \mathbb{Q}=...
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How is the behaviour of a deformation retract under a fibration? [duplicate]
Let $p:E \rightarrow B$ a fibration and take $A\subset B$ a deformation retract of B. Is it true that $p^{-1}(A)$ is a deformation retract of E?
By deformation retract I mean the weaker definition.
I'...
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Implementation of the nerve of a category in GAP
I am trying to compute the second cohomology group of the coset poset $$\mathcal{C}_G\mathcal{F}=\{gA\mid g\in G,A\in\mathcal{F}\}$$ for a finite group $G$ and a family $\mathcal{F}$ of subgroups of $...
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Is taking Freyd envelopes adjoint to taking stable module categories?
Let $T$ be an (idempotent-complete) triangulated category. Then the Freyd envelope $mod(T)$ is an abelian category, the universal recipient of a homological functor $T \to mod(T)$. The Freyd envelope ...
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Plus construction on Simplicial Sets?
I had asked this question in Math StackExchange a few days ago, but didn't get any answers. I believe its more suitable to be asked here.
Write $\mathsf{sSet}$ for the category of simplicial sets and $...
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Is the Whitehead bracket $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$ injective?
Let $X$ and $Y$ be finite CW-complexes and $p,q\geq 2$. The Whitehead bracket induces a homomorphism $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$, $\alpha\otimes \beta\mapsto [\alpha,\beta]$....
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Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection
I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic ...
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Infinite loop space as an endofunctor of compactly generated weak hausdorff topological spaces?
I am trying to see whether it is possible to define smash product of infinite loop spaces using the space $S^{\infty}$.
Let C be the category of compactly generated weak Hausdorff topological spaces. ...
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Reference request for equivalences between different models of lax limits
There are several models for lax limits of model categories/ $\infty$-categories in the literature. For example, within the realm of $\infty$-categories one can construct them using coCartesian ...
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Hypercube of chain complexes as functor from (Δ^1 )^n to ∞-category of chain complexes
A hypercube of chain complexes consists of $\mathbb{Z}$-graded vector spaces $C_\epsilon$ for $\epsilon\in\{0,1\}^n$ and maps $D_{\epsilon,\epsilon^\prime}:C_{\epsilon}\to C_{\epsilon^\prime}$ for $\...
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Bⁿ and coherence
I understand that an internal abelian group in CW-complexes (while it is not the most extensive structure for which this can be done) produces a classifying space which itself has the structure of an ...
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Duality in Spc with ∧ and [-,-]
I am thinking about two duality theorems for H-spaces and their actions.
By H-space is meant a commutative monoid in the derived (homotopy) category of based connected CW-complexes. We can consider ...
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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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Does the forgetful functor $F:\mathrm{CAlg}\to\mathrm{Alg}^{(1)}$ sending $E_\infty$-ring spectra to $E_1$-ring spectra preserve limits and colimits?
In remark 7.1.0.4. of Lurie's Higher Algebra, the sequence $(E_n^{\otimes})_{0\leq n\leq\infty}$ of $\infty$-operads induces forgetful functors for the sequence of categories $(\operatorname{Alg}^{(n)}...
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Can a phantom map have finite cofiber?
Let $f : X \to Y$ be a nonzero phantom map between spectra. Can the cofiber of $f$ be a finite spectrum?
Recall that a map $f$ is said to be phantom if $f \circ i = 0$ whenever $i : F \to X$ is a map ...
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Is the mapping cylinder a replacement for morphism by cofibration in model categories?
Let $M$ be a model category, consider a very good cylinder object $X \coprod X \to X \times I \overset{\operatorname{pr}}{\to} X$ (here $X \times I$ is just a notation, no object $I$ is implied), that ...
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Properties of colim Ωⁿ Σⁿ X
I am thinking about the paper of Gaunce Lewis Jr. showing the incompatibility of a certain five desirable properties of spectra. This paper makes me curious about the properties of the endofunctor $Q: ...
user30211
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Local-to-global philosophy for crossed modules
In the survey Groupoids and crossed objects in algebraic topology Ronald Brown made after Corollary 5.17 (p 30) an very interesting remark I not fully understand. He stated that this
Corollary 5.17 ...
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A question on $BP$ and $E_\infty$ models for ring spectrums
I am a beginner in this field. My question is
(1) Is the existence of $E_\infty$ ring structure not closed under weak equivalence of ring spectra?
(2) If (1) is true, what is the risk of replacing a ...
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Symmetric-monoidal-associative smash product up to homotopy
I am thinking about sequential spectra. I am trying to figure out if the smash product here is symmetric monoidal associative up to homotopy. See definition 3.16 in the above.
Recall that a sequential ...
user30211
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Semi-simple algebras over operads
I believe people thought about this questions, however I couldn't find any reference. I appreciate if someone could direct me to some detailed discussions about it.
The categories of associative ...
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Homotopy fixed point spectral sequence for cooperation algebra
For a finite group $G$ acting on a spectrum $X$, there is a homotopy fixed point spectral sequence calculating $X^{hG}$ with signature
$$E_2^{p, q} = H^p(G, \pi_q X) \implies \pi_{q-p}(X^{hG}).$$
...
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The homotopy inverse on Quillen's $S^{-1}S$ construction
Suppose $S$ is a symmetric monoidal groupoid. Take Quillen and Grayson's $S^{-1}S$-construction, which is a symmetric monoidal category with objects given by pairs $(m,n)$ and maps given by ...
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Connection on relative topological periodic cyclic homology
I have been looking Bhatt-Morrow-Scholze's paper:
https://arxiv.org/pdf/1802.03261.pdf
and came to a naive question. Let $C$ be a dg-category (with assumptions?) over $\mathbb{F}_p[[z]]$ and view this ...
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