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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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
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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
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6 votes
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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
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141 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
239 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
3 votes
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60 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
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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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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
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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
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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
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7 votes
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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
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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
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4 votes
1 answer
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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
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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
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62 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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8 votes
1 answer
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On the connections between condensed mathematics and homotopy theory

I have a few questions, but they're not properly formulated just yet, but they stem from a few simple facts : In homotopy theory, the homotopy hypothesis postulates that topological spaces (up to ...
t_kln's user avatar
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2 answers
151 views

Reference for choosing a path lifting function?

I recall having seen discussion of a Hurewicz or Serre fibration equipped with a chosen path lifting function. Citation??
jim stasheff's user avatar
3 votes
0 answers
88 views

Confusion about signs in the definition of an $A_\infty$-algebra

We are trying to understand the definition of $A_\infty$-algebras. But we are puzzled by what appear to be two different sign conventions (and we cannot figure out how these two are equivalent) We see ...
ainfg's user avatar
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Examples of when $X$ is homotopy equivalent to $X\times X$

I was thinking about this question the other day: When is a topological space $X$ homotopy equivalent to $X\times X$ (with the product topology)? This is essentially a cross-post of this MSE question.....
pyridoxal_trigeminus's user avatar
2 votes
0 answers
128 views

Tensor product of objectwise weak homotopy equivalences of $\mathcal{M}$-spaces

I consider the enriched category $[\mathcal{M}^{op},\mathrm{Top}]$ of enriched functors (I call them $\mathcal{M}$-spaces) from the enriched small category $\mathcal{M}^{op}$ to the enriched category $...
Philippe Gaucher's user avatar
3 votes
1 answer
187 views

On infinity-morphisms between algebras over algebraic operads

I posted this question in the "Mathematics" stack exchange, but it hasn't got much attention... I hope it will get more here. Let $P$ be a Koszul operad. In the book of Loday-Vallette "...
groupoid's user avatar
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Explicit CW-complex replacement of the space of reparametrization maps

Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-...
Philippe Gaucher's user avatar
1 vote
0 answers
104 views

Lengths and additive invariants which preserve positivity

The length of a module is well-known to be an additive invariant of finite-length modules. That is, if $R$ is a ring and $Art(R)$ its category of finite-length modules, then $length : Ob (Art(R)) \to \...
Tim Campion's user avatar
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0 votes
0 answers
100 views

Basis of Lambda algebra for a programmer

First of all, I'm not a specialist in alg. top., but I try to apply computational math to it, so if I'm wrong in something you'd be doing a better thing explaining it to me instead of blaming me :) ...
Dmitry Vilensky's user avatar
5 votes
1 answer
252 views

Do finitely presentable $\infty$-groupoids precisely correspond to the finite cell complexes?

In the Higher Topos Theory, Example 1.2.14.2 says “finitely presentable $\infty$-groupoids correspond precisely to the finite cell complexes” But, for example, $K(\mathbb{Z}, 2)$ is seems finitely ...
Arshak Aivazian's user avatar
2 votes
0 answers
65 views

Examples of tensor-triangulated categories not satisfying the local-to-global principle

From now on, we will consider only rigid-compactly generated tensor-triangulated categories. Let $(\mathcal{T}, \otimes, 1)$ be one of these categories, it is known that the thick tensor ideals of ...
N.B.'s user avatar
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2 votes
1 answer
255 views

Filtered homotopy colimits of spectra

Let $\mathcal{I}: \mathbb{N} \to \operatorname{Sp}$ be a diagram in the infinity category of spectra. Let $\pi_0(\mathcal{I})$ denote the corresponding $1$-categorical diagram (i.e. compose $\mathcal{...
Laurent Cote's user avatar
3 votes
1 answer
228 views

1D topological defects in $d>3$ spatial dimensions

I am trying to construct a 1D topological defect solution in 4 spatial dimensions, i.e., a solution to some PDE (likely the equations of motion of some Lagrangian) on $\mathbb{R}^{4}$ which is ...
math_lover's user avatar
11 votes
1 answer
399 views

Is an exponentiable fibration with contractible fibers a homotopy equivalence?

Question: Let $p : E \to B$ be an exponentiable functor of $\infty$-categories. Suppose that for every $b \in B$, the geometric realization of the fiber $|p^{-1}(b)|$ is contractible. Then does $p$ ...
Tim Campion's user avatar
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2 votes
1 answer
80 views

Reference request-Natural equivalence detected pointwise for complete Segal spaces

I am looking for a reference for the following elementary assertion on complete Segal spaces: Let $A$ be a bisimplicial set and let $W$ be a complete Segal space. A morphism of $W^A$ is an ...
Ken's user avatar
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2 votes
0 answers
114 views

Homotopy type of a 3-manifold produced via Dehn surgery?

My apologizes if this is a fairly elementary question, I am still a novice when it comes to 3-manifold topology. I am wondering the following: by Kirby calculus, we know that two links (say in $S^{3}$ ...
Elliot's user avatar
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4 votes
2 answers
278 views

On a generalized homotopy transfer theorem

In the book of Loday and Vallette "Algebraic Operads" a necessary condition for the Homotopy Transfer Theorem is that the starting operad is Koszul. I am interested in a generalization of ...
groupoid's user avatar
  • 217
13 votes
2 answers
486 views

How many automorphisms are there of the category of filtered spectra?

Dold-Kan type theorems tell us that lots of categories are Morita-equivalent to the simplex category $\Delta$. In other words, there are a lot of stable $\infty$-categories which are secretly ...
Tim Campion's user avatar
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1 vote
1 answer
144 views

Compact locus in (ordered) configuration spaces

Let $\mathit{Conf}_n(\mathbb{R}^2)$ be the configuration space of $n$ ordered distinct points in the plane. I'd like to know if the topological subspace $C_n$ consisting of points $(p_1,...,p_n)$ with ...
John Gunnar Carlsson's user avatar
2 votes
0 answers
109 views

Homotopy equivalence of chain complexes from subcomplexes and quotient complexes

Let $C_1$ be a finite-dimensional chain complex over $\mathbb{C}$ coefficients. Let $S_i$ be a subcomplex of $C_1$ and let $Q_1$ be the quotient complex. Suppose $S_1$ and $Q_1$ are chain homotopy ...
Faniel's user avatar
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3 votes
2 answers
271 views

Cut a homotopy in two via a "frontier"

Consider a space $G$ obtained by glueing two disjoint cobordisms (the fact that they are might be irrelevant, assume they are topological spaces at first) $L$ and $R$ on a common boundary $C$. (...
Valentin Maestracci 's user avatar
9 votes
1 answer
430 views

Is there a shape-independent definition of (∞,1)-categories?

For all definitions of $\infty$-categories I am aware of, an $(\infty,1)$-category is defined via reference to some shape, be it simplices in a form of a quasi-category or a cubical analogue of a ...
lemmanade's user avatar
2 votes
1 answer
115 views

Homotopy groups of $K(n)$-local $E_n$-modules are $L$-complete

Let $E_n$ be the $n$-th Morava $E$-theory and let $K(n)$ denote the $n$-th Morava $K$-theory. Question: If $M$ is a $K(n)$-local $E_n$-module, then are the homotopy groups $\pi_*(M)$ $L$-complete? (...
happymath's user avatar
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6 votes
1 answer
379 views

A possible alternative model for $\infty$-groupoids

I've been studying homotopy theory on myself for quite some time now, and it is to my understanding that there's still no generally accepted definition for $\infty$-groupoids. The closest to this is ...
XiaohuWang's user avatar
4 votes
0 answers
428 views

An $E_{\infty}$-algebra is a $C_{\infty}$-algebra?

Past this question in MO have raised the following questions for me. Question In characteristic $0$, it is well-known that a Kadeishvili‘s $C_{\infty}$-algebra is an $E_{\infty}$-algebra. However, do ...
Walterfield's user avatar
2 votes
2 answers
241 views

The complex $K$-theory of the Thom spectrum $MU$

The Atiyah-Hirzebruch spectral sequence is a strong computational tool that yields several interesting computation in (co)homology. I want to know whether $K_\ast(MU)$ and $K^\ast(MU)$ have been ...
Plius's user avatar
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6 votes
1 answer
225 views

Mapping spaces in complete Segal spaces and quasi-categories

Complete Segal spaces and quasi-categories are two common models for the theory of $(\infty,1)$-categories, and both are equipped with a natural notion of hom spaces. For complete Segal spaces, which ...
ChrisLazda's user avatar
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5 votes
1 answer
149 views

Simplicial objects in quasicategory which come from homotopy coherent nerve

Let $\mathcal{C}$ be a simplicially enriched category whose Hom-objects are all Kan complexes. Denote by $N\mathcal{C}$ the homotopy-coherent nerve of $\mathcal{C}$, which is a quasicategory. Suppose ...
K. Strong's user avatar
  • 335
2 votes
0 answers
306 views

Square-zero extensions mod $p^n$

$\DeclareMathOperator\LMod{LMod}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Sp{Sp}$A square-zero extensions of rings is, conceptually, a map of rings $R \to A$ such that any two elements in the ...
Mori B.'s user avatar
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3 votes
0 answers
113 views

Does symmetric product functor preserve fibrations?

I know that the symmetric product is a functor, cf: https://en.wikipedia.org/wiki/Symmetric_product_(topology)#Functioriality. My question is, does it preserve fibrations in the category of ...
JE2912's user avatar
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5 votes
0 answers
121 views

Division of fibration by $\Sigma_{n}$ gives Serre fibration

This is related to a question posted on StackExchange: https://math.stackexchange.com/questions/4776877/left-divisor-of-a-fibration-by-compact-lie-group-is-a-fibration. The question there had received ...
TopologyStudent's user avatar
2 votes
0 answers
181 views

The derived category of $p$-complete abelian groups is comonadic over the derived category of $\mathbb F_p$-vector spaces?

$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ext{Ext}$Let $p$ be a prime. The adjunction $$\mathbb F_p \otimes_\mathbb{Z} (-) : \Mod(\mathbb Z) \rightleftarrows \Mod(\mathbb F_p) : U $$ descends ...
Tim Campion's user avatar
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0 votes
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A question on the proof of pullback bundles by homotopic maps are isomorphic in Prof. Ralph Cohen notes

Here is a question about proving the pullback bundles by homotopic maps are isomorphic in Prof. Ralph Cohen notes Bundles, Homotopy, and Manifolds. The proof is in page 73 of the notes. For me, ...
Ho Man-Ho's user avatar
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6 votes
1 answer
347 views

How to differentiate natural transformations?

Let $G_{\bullet} = (G_1 \rightrightarrows G_0)$ and $H_{\bullet} = (H_1 \rightrightarrows H_0)$ be Lie groupoids and $\varphi_\bullet, \psi_\bullet : G_\bullet \to H_\bullet$ be Lie groupoid morphisms,...
Felix Lungu's user avatar
2 votes
0 answers
58 views

Coequalizers and pullbacks in $\infty$-topoi

In an $\infty$-topos, suppose we have two cartesian diagrams of the form $$ \require{AMScd} \begin{CD} \overline{A} @>>> \overline{B} \\ @VVV @VVV \\ A @>>> B . \end{CD} $$ Let $$ \...
grass man's user avatar

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