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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Motivation and potential applications of spectral algebraic geometry
Nowadays there is a lot of talk about derived algebraic geometry, but not so much about the related subject of spectral algebraic geometry.
Now I'm curious what future is there for spectral algebraic ...
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How does it End?
A recent project has forced my colleague and me to take a rather abstract approach to dynamical systems, and the following definition arose naturally in that context.
Let $\mathcal{C}$ be a category. ...
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Maps out of Eilenberg-Mac Lane Spaces
Is anything known about the maps out of an Eilenberg Mac-lane Space $K(G,n)$?
Obviously I'm interested in extensions of Miller's resolution of the Sullivan conjecture, that $Map_*(K(G,1),X)\simeq\ast$...
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Why does elliptic cohomology fail to be unique up to contractible choice?
It is often stated that the derived moduli stack of oriented elliptic curves $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ is the unique lift of the classical moduli stack of elliptic curves satisfying some ...
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Is the J homomorphism compatible with the EHP sequence?
The EHP sequence consists of maps $S^n\stackrel{E}{\to} \Omega S^{n+1} \stackrel{H}{\to} \Omega S^{2n+1}$. This is a homotopy fibration sequence if $n$ is odd. For even $n$ it is a homopy fibration ...
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obstruction theories in algebraic geometry
I'd like to know about the history of obstruction theories in algebraic geometry, as well as the relationship with concepts of the same name in topology. I would also like to know where obstruction ...
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Is algebraic $K$-theory a motivic spectrum?
I've received conflicting messages on this point -- on the one hand, I've been told that "forming a natural home for algebraic $K$-theory" was one motivation for the development of motivic homotopy ...
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Does derived algebraic geometry allow us to take quotients with reckless abandon?
So one of the major problems with the categories of schemes and algebraic spaces is that the "correct quotients" are oftentimes not schemes or algebraic spaces. The way I've seen this sort of thing ...
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Morphism from a surface group to a symmetric group, lifted to the braid group
Let $\Sigma_g$ be the fundamental group of the closed orientable surface of genus $g\ge 2$; let $B_n$ be the braid group on $n\ge 3$ braids; let $S_n$ be the symmetric group on $n$ letters; let $p:B_n\...
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How to stop worrying about enriched categories?
Recently I realized that ordinary category theory is not a suitable language for a big portion of the math I'm having a hard time with these days. One thing in common to all my examples is that they ...
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If $X\times Y$ is homotopy equivalent to a finite-dimensional CW Complex, are $X$ and $Y$ as well?
Is there a space $X$ that is not homotopy equivalent to a finite-dimensional CW complex for which there exists a space $Y$ such that the product space $X\times Y$ is homotopy equivalent to a finite-...
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Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets?
Many topologists express a clear preference for working with CW complexes instead of simplicial sets.
One of the reasons is that the cellular chain complex of a CW complex is often easier to work ...
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The first unstable homotopy group of $Sp(n)$
Thanks to the fibrations
\begin{align*}
SO(n) \to SO(n+1) &\to S^n\\
SU(n) \to SU(n+1) &\to S^{2n+1}\\
Sp(n) \to Sp(n+1) &\to S^{4n+3}
\end{align*}
we know that
\begin{align*}
\pi_i(SO(...
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Mark Hovey's open problems in the theory of model categories
Mark Hovey maintains a list of open problems in model category theory. I think this list is quite old, and I don't know if Hovey is still updating it or not.
My question is:
i) which of the 13 ...
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Is there an additive model of the stable homotopy category?
$\DeclareMathOperator\Ho{Ho}$Is there a model category $C$ on an additive category such that its homotopy category $\Ho(C)$ is the stable homotopy category of spectra and the additive structure on $\...
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Are non-empty finite sets a Grothendieck test category?
A "test category" is a certain kind of small category $A$ which turns out to have the following property: the category $\widehat{A}$ of presheaves of sets on $A$ admits a model category structure, ...
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What is the homotopy theory of categories?
I've heard that Grothendieck, in his letter "Pursuing Stacks," wanted to find alternative models for the classical homotopy category of CW complexes and continuous maps (up to homotopy), and one of ...
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Sheaves of complexes and complexes of sheaves
Let A be an abelian category, and X a topological space.
There are two ways one could try to construct some oo-category of sheaves on X from this data:
Consider the category $Sh(X,A)$ of sheaves on ...
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homotopy type of connected Lie groups
Is there a simple proof (short and low-tech) of the following fact:
(E. Cartan) A connected real Lie group $G$ is diffeomorphic (as a manifold) to
$K\times\mathbb{R}^n$ where $K$ is a maximal ...
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Does Waldhausen K-theory detect homotopy type?
Recall that $A(X)$, the K-theory of a connected, pointed space X, is defined as the K-theory spectrum of the ring spectrum $\Sigma^\infty_+ \Omega X$ (or via a plethora of alternative definitions). Is ...
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Stable homotopy type theory?
This is a combined question, strictly speaking I am asking three questions concerning, respectively, homotopy type theory, stable homotopy theory and Yetter-Drinfeld modules. But I believe in the ...
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Is there a generalization of homotopy groups to fractional dimensions
Does there exist a reasonable candidate for such an object as $\pi_{\frac12}(X)$?
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Étale homotopy type of non-archimedean analytic spaces
The following is likely all obvious to the experts. But since the field looks tricky to an outsider, maybe I may be excused for asking anyway.
I am wondering about basic facts of what would naturally ...
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The cell structure of Thom spectra
I would like to understand the cell structure of integrally oriented Thom spectra. A Thom spectrum over a space $X$ is something you can build from a stable spherical bundle, which is classified by a ...
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Super-cobordisms
One can construct the $d$-dimensional bordism category by declaring the objects to be the $(d-1)$-dimensional compact manifolds without boundary and the morphisms the $d$-dimensional bordisms between ...
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Distinct manifolds with the same configuration spaces?
For a space $X$, let $C_k X$ denote the space of configurations of $k$ distinct unordered points in $X$.
What is an example of a pair of smooth manifolds $M$ and $N$ that are not homeomorphic but ...
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How many model categories have the same weak equivalences?
There are many situations which arise where one might consider different Model categories with the same underlying category. For example in (left) Bousfield localization you start with a model ...
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Table of (integral) cohomology groups of K(Z,n)
Can I find somewhere a table of the (first few) cohomology groups of $K(\mathbb{Z},n)$ with integer coefficients?
It seems like a natural counterpart to the table of the homotopy groups of spheres, ...
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Acyclic models via model categories?
Recall the acyclic models theorem: given two functors $F, G$ from a "category $\mathcal{C}$ with models $M$" to the category of chain complexes of modules over a ring $R$, a natural transformation $...
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What is the group completion of finite sets with respect to cartesian product?
Let $\Sigma_+$ be the groupoid of finite pointed sets. The Barratt-Priddy-Quillen theorem tells us that the group completion of $\Sigma_+$ with respect to the symmetric monoidal structure given by ...
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Does the holonomy map define a homomorphism $\pi_k(X)\to\pi_{k-1}(Hol(\nabla))$?
Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...
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Homotopic version of Freyd's AT category observations
Freyd was the first to formalize a striking comparison between abelian categories and topoi, showing that their exactness properties can be jointly captured by the axioms of AT categories, and the ...
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Computing homotopies
Oftentimes, in the standard algebraic topology books (May, Switzer, Whithead, for instance), there are tricky little proofs that depend on proving that two maps are homotopic. This is comparable to ...
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Difference between represented and singular cohomology?
Ordinary cohomology on CW complexes is determined by the coefficients. There are (more than) two nice ways to define cohomology for non-CW-complexes: either by singular cohomology or
by defining $\...
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Homotopy pullbacks and homotopy pushouts
I have a good grasp of ordinary pullbacks and pushouts; in particular, there are many categorical constructions that can be seen as special cases: e.g., equalizers/coequalizers, kernerls/cokernels, ...
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Does the cohomology ring of a simply-connected space X determine the cohomology groups of ΩX?
One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but ...
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The definition of homotopy in algebraic topology
In this post, let $I=[0,1]$.
Something about the definition of homotopy in algebraic topology (and in particular in the study of the fundamental group) always puzzled me. Most books on the ...
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Are there prominent examples of operads in schemes?
There is an abundance of examples of operads in topological spaces, chain complexes, and simplicial sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even ...
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What are finite homotopy types?
Starting from an ordinary 1-categorical point of view, there are various obvious candidate definitions for ‘finite homotopy type’:
The homotopy type of a simplicial set that has only finitely many ...
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Is the 4-line of the E_2 term of the classical Adams spectral sequence known?
In other words:
What is $\mathrm{Ext}_{\mathcal{A}}^{4,t}(\mathbb{Z}/2,\mathbb{Z}/2)$?
If the 4-line is not known, how much is known about it?
Here, $\mathcal{A}$ is the 2-primary Steenrod ...
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Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?
With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$?
Here, by "trivial examples" ...
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Vector bundles on vector bundles
Are all vector bundles on a given vector bundle the pull back of a vector bundle on the base?
In more detail: let $X$ be a space and $p:E\rightarrow X$ a vector bundle over $X$. Let $\iota: X \...
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Homotopy equivalent Postnikov sections but not homotopy equivalent
Two pointed, connected CW complexes with the same homotopy groups need not be homotopy equivalent (Are there two non-homotopy equivalent spaces with equal homotopy groups?). Moreover, having the same ...
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What is classified by generalised Eilenberg MacLane spaces?
Given an abelian group $A$, the Eilenberg MacLane spaces $K(A,n)$ represent the the nth cohomology group in $A$.
In a similar vein, given an arbitrary group $G$ and a space $X$, maps to the ...
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A "universally non Hypercomplete" $\infty$-topos via Goodwillie calculus?
My question is :
Is there a classifying $\infty$-topos for $\infty$-connected objects ? Does this $\infty$-topos has a nice description (as an $\infty$-category ) ?
What I mean by $\infty$-connected ...
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What are the fibrant objects in the injective model structure?
If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial ...
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When does the free loop space fibration split?
This question is a repost from stack.exchange. It didn't get a lot of attention there. Perhaps it is badly written (or silly?). If so, I'd be happy to get comments/suggestions about that.
Let $X$ be ...
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Are Chern classes well defined up to contractible choice?
The Chern classes are, by definition, cohomology classes. And
cocycle representatives of the Chern classes are not unique.
But it might be the case that cocycle representatives of the Chern classes ...
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Which cohomology classes are detected by tori?
Given a space $X$, I am looking for a characterization of classes $\alpha \in H^n(X;\bf Q)$ such that there is a map $f\colon T^n \to X$ so that $f^{\ast} \alpha$ pairs non-trivially against the ...
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What is obstructing two stably-isomorphic vector bundles from being isomorphic?
The specific situation is the following:
Let $n>0$ be a natural number, let $X$ be a finite CW complex of dimension $n$, and let $\xi_0,\xi_1$ be oriented real vector bundles of rank $n$ over $X$ ...
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