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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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. ...
Ronald J. Zallman's user avatar
6 votes
0 answers
104 views

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 ...
happymath's user avatar
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2 votes
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106 views

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 $\...
Faniel's user avatar
  • 593
3 votes
1 answer
143 views

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 ...
Ronald J. Zallman's user avatar
1 vote
0 answers
119 views

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 ...
Ronald J. Zallman's user avatar
7 votes
1 answer
402 views

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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4 votes
1 answer
149 views

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)}...
Miso's user avatar
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3 votes
1 answer
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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 ...
Tim Campion's user avatar
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2 votes
2 answers
201 views

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 ...
Arshak Aivazian's user avatar
1 vote
0 answers
201 views

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: ...
Ronald J. Zallman's user avatar
2 votes
0 answers
137 views

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 ...
user267839's user avatar
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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 ...
Miso's user avatar
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1 vote
1 answer
215 views

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 ...
Ronald J. Zallman's user avatar
7 votes
1 answer
105 views

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 ...
Li Guanyu's user avatar
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0 answers
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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 $$E_2^{p, q} = H^p(G, \pi_q X) \implies \pi_{q-p}(X^{hG}).$$ ...
categorically_stupid's user avatar
7 votes
0 answers
107 views

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 ...
Georg Lehner's user avatar
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2 votes
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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 ...
Daniel Pomerleano's user avatar
2 votes
0 answers
192 views

Using the Dold-Thom Theorem to define \'etale cohomology

For reasonable spaces $X$, the Dold-Thom Theorem states that $\pi_i(SP(X)) \cong \tilde{H}_i(X)$ where $SP(X) = \bigsqcup_i \mathrm{Sym}^i(X)$. There is a purely algebro-geometric realization of this ...
Asvin's user avatar
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4 votes
0 answers
231 views

Homotopy group of maps into S^3 using its Lie group multiplication to define the group operation

The Bruschlinsky group of maps of a space X into S1 up to homotopy, using the multiplication on S1, is well-known to equal the first cohomology group of X (at least assuming X is a reasonably nice ...
Daniel Asimov's user avatar
2 votes
0 answers
135 views

A possible generalization of "homotopy" to study group actions of various kinds

This is a naive question about abstract homotopy theory by someone who knows nothing about it, except that it involves some generalization of the notion of "homotopy". If we think of $O(n)$ ...
semisimpleton's user avatar
14 votes
1 answer
829 views

What is $\pi_{23}(S^2)$?

The first $22$ homotopy groups of the $2$-sphere were worked out by Toda in 1962, but I cannot find any results extending that to any higher homotopy groups of $S^2$. Are any more of these groups ...
Joe Shipman's user avatar
6 votes
1 answer
181 views

Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations

Let $\mathrm{Top}$ be the category of all topological spaces and continuous maps. The Quillen model structure on $\mathrm{Top}$ has weak equvalences $W = \{ \text{weak homotopy equivalences} \}$, ...
mathmo's user avatar
  • 161
13 votes
2 answers
918 views

Categories on which one can determine all model structures?

Famously, there are exactly nine model structures on the category of sets, which are detailed here. In this case, one can exhaustively determine all six weak factorization systems and then see which ...
Tim Campion's user avatar
  • 60.2k
14 votes
3 answers
575 views

Strøm model structures on the category of simplicial sets

Let $X,Y$ be simplicial sets. A simplicial homotopy is a simplicial map of the form $h:X\times\Delta^1\rightarrow Y$. There are two distinguished maps $$ in_0:X\cong X\times\Delta^0\xrightarrow{1\...
Tyrone's user avatar
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5 votes
1 answer
283 views

Are there “nice” explicit representations of infinite order elements in $\pi_{4n-1}(S^{2n})$

Motivation: The representation is very clean in the Hopf fibration case (2n=2,4,8). These maps are known since Serre's 1951 paper. There are clean generators for the only other infinite case $\pi_n(S^...
David Lehavi's user avatar
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3 votes
1 answer
130 views

Linearity of topological periodic cyclic homology

Let $A$ be an $E_\infty$ ring spectrum, $B$ a ring spectrum. Then if I understand correctly, $TP(A)$ is a ring spectrum by the lax monoidal property of $TP$. Suppose there is a map of ringed spectra ...
onefishtwofish's user avatar
4 votes
1 answer
324 views

How to calculate $\mathrm{TP}(\mathbb{F}_p[t])$?

$\DeclareMathOperator\TP{TP}$I am trying to learn about topological periodic cyclic homology following the notes: https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Lectures.pdf https://...
onefishtwofish's user avatar
5 votes
1 answer
320 views

Non-triviality of Whitehead products in wedges of CW-complexes

Suppose $X$ and $Y$ are finite, simply connected, based CW-complexes and $m,n\geq 2$. If $a\in \pi_m(X)$ and $b\in \pi_n(Y)$, then one can regard these as elements of the homotopy groups of $X\vee Y$. ...
J.K.T.'s user avatar
  • 395
15 votes
1 answer
731 views

If homotopy groups of spaces are identical, then stable ones are also identical?

Is it true that if pointed spaces $X, Y$ have the same homotopy groups $\pi_n(X) \cong \pi_n(Y)$, then they have the same stable homotopy groups $\pi^S_n(X) \cong \pi ^S_n(Y)$? In particular, is this ...
Arshak Aivazian's user avatar
2 votes
0 answers
162 views

Triviality of map $(\Sigma \theta)^*$

We know that there is a cofibration sequence $$S^{4n+1}\xrightarrow{\theta}\Sigma^{4m-1} Q_{n-m} \rightarrow \Sigma^{4m-1} Q_{n-m+1} \rightarrow S^{4n+2}\xrightarrow{\Sigma\theta}\Sigma^{4m} Q_{n-m}.$$...
Sajjad Mohammadi's user avatar
9 votes
1 answer
588 views

Homotopy groups of finite CW complex finitely generated as Lie algebra

This is probably a well-known question, but I haven't found the answer on MO or MSE. It is well-known that the homotopy groups of a finite CW complex $X$ need not be finitely presented, even as $\...
R. van Dobben de Bruyn's user avatar
4 votes
1 answer
338 views

Two spectral sequences arising from a simplicial spectrum

Let $X_\bullet$ be a simplicial spectrum, and let $X = |X_\bullet|$ be the geometric realization. Let's assume each $X_n$ is connective. From this situation, we can form two filtrations on $X$: the ...
Brian Shin's user avatar
10 votes
1 answer
403 views

Where does the definition of ($\infty$-)groupoid cardinality come from?

The cardinality of a finite set $X$ is the number of its elements. Once you know that, you would define the groupoid cardinality of a $\infty$-groupoid $X$ as the quantity $$\lvert X\rvert := \sum_{[x]...
Matthew Niemiro's user avatar
7 votes
0 answers
172 views

Complex cobordism and integrable systems

In Jack Morava's paper On the complex cobordism ring as a Fock representation, it was remarked right at the beginning that complex cobordism may play a role in the theory of integrable systems. In ...
user1271629's user avatar
4 votes
0 answers
76 views

For a map $x: S^0 \to X$, $J(X,x) \otimes Y = 0$ iff $x \otimes 1_Y$ is nilpotent?

Let $X$ be a spectrum, and let $x : S^0 \to X$ be a map. Let the stable James construction $J(X,x)$ denote the free $E_1$ ring on the $E_0$-ring $(X,x)$. It is computed as the colimit of the $\Delta^{...
Tim Campion's user avatar
  • 60.2k
8 votes
0 answers
129 views

The James and Morse filtrations of homotopy groups

Denote by $JX$ the James construction on a path connected, well-pointed space $X$. This space is filtered by subspaces $X=J_1X\subseteq\dots\subseteq J_nX\subseteq\dots\subseteq JX$ and is the domain ...
Tyrone's user avatar
  • 4,961
4 votes
1 answer
229 views

On the initiality of the inclusion from the simplex category to the paracycle category

Thm B.3 of Nikolaus and Scholze shows that the natural inclusion $\Delta \to \Lambda_\infty$, from the simplex category to the paracycle category, is an initial functor, i.e. satisfies the hypotheses ...
Tim Campion's user avatar
  • 60.2k
6 votes
1 answer
406 views

Does $\pi_1(H)=0\Rightarrow \pi_3(G/H)=0$ for a simple and simply connected Lie group $G$?

$\DeclareMathOperator\SU{SU}$Let $G$ be a simple and simply-connected Lie group and $H\neq 1$ be a simple and simply connected subgroup, is it true that $\pi_3(G/H)=0$? If not, what is a counter-...
Davi Costa's user avatar
7 votes
0 answers
136 views

Explicit framed null bordism realizing $\eta\nu =0$ in stable homotopy group of spheres

There are many standard results in the stable homotopy group of spheres (or equivalently framed bordism groups), about which I would like to acquire better geometric understanding. For example I ...
Yuji Tachikawa's user avatar
4 votes
1 answer
238 views

Two $E_\infty$ structures on infinite matrices

Let $O$ be the infinite orthogonal group. By taking a colimit of the diagram of topological groups $O(1) \to O(2) \to O(3) \to \ldots$, we know $O$ has a continuous group operation given by matrix ...
guest313131's user avatar
2 votes
0 answers
55 views

Example of a factorisation of functors $F = HK$ for which the Kan extension of $F$ along $K$ is not $H$

I was reading Emily Riehl's book: Categorical Homotopy theory, and I encountered exercise 1.1.3: Exercise 1.1.3: Construct a toy example to illustrate that if $F$ factors through $K$ along some ...
julio_es_sui_glace's user avatar
6 votes
1 answer
220 views

Does a complex-oriented $E_1$ ring spectrum (not assumed to have graded-commutative homotopy groups) receive a map from $MU$?

It's well-known that complex cobordism $MU^\ast$ is universal among complex-oriented associative, graded-commutative cohomology theories $E$. This means that if $E$ is a multiplicative cohomology ...
Tim Campion's user avatar
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3 votes
2 answers
407 views

A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $

Consider a manifold $ N $ defined as follows $$ N=\{n\otimes n-m\otimes m:n,m\in S^2,\quad(n,m)=0\}\subset M^{3\times 3}, $$ where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ ...
Luis Yanka Annalisc's user avatar
23 votes
3 answers
2k views

What are some toy models for the stable homotopy groups of spheres?

The graded ring $\pi_\ast^s$ of stable homotopy groups of spheres is a horrible ring. It is non-Noetherian, and nilpotent torsion outside of degree zero. Question: What are some "toy models" ...
Tim Campion's user avatar
  • 60.2k
2 votes
0 answers
62 views

What are the epis, monos, and extensions in the Freyd Envelope of a triangulated category?

Let $\mathcal T$ be a triangulated category (or homotopy category of a stable $\infty$-category). Recall that the Freyd envelope of $\mathcal T$ is an abelian category $\mathcal A$ which is ...
Tim Campion's user avatar
  • 60.2k
7 votes
0 answers
261 views

Homotopy theory of differential objects

In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...
ಠ_ಠ's user avatar
  • 5,875
4 votes
0 answers
91 views

What is the Goldie dimension of the ring of stable stems?

Let $p$ be a prime, and let $\pi_\ast^{(p)}$ be the ring of stable homotopy groups of spheres localized at the prime $p$. This is a nonnegatively-graded-commutative ring with $\mathbb Z_{(p)}$ in ...
Tim Campion's user avatar
  • 60.2k
4 votes
1 answer
156 views

The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$

$\DeclareMathOperator\BGL{BGL}$In the paper, 'Two-primary Algebraic $K$-theory of rings of integers in number fields', Rognes and Weibel compute the $2$-torsion part in the algebraic $K$-theory of the ...
atinag's user avatar
  • 43
1 vote
0 answers
191 views

Kan extensions in Grothendieck school

Considering both the ubiquity of Kan extensions in category theory (as MacLane stated, 'The notion of Kan extensions subsumes all the other fundamental concepts of category theory.'), its early ...
user234212323's user avatar
2 votes
0 answers
59 views

Framed bordism and string bordism in 3-dimensions vs topological modular form

In simple colloquial terms, how are the framed bordism and string bordism in 3-dimensions related to the study of the theory of topological modular form TMF? I want to know some simple derivable ...
wonderich's user avatar
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