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.
123 questions from the last 365 days
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What are all the (orthogonal) factorization systems on infinity groupoids?
As long as I didn't forget to verify some condition, there are exactly four (orthogonal) factorizations on Set:
The two trivial ones (any, iso) and (iso, any).
The (surjection, injection), aka image ...
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How to understand the homotopy groups of the realization of a simplicial space?
I am a beginner in simplicial homotopy theory. My knowledge in this area mainly comes from reading the first two chapters of Simplicial Homotopy Theory by Paul G. Goerss and John F. Jardine, namely ...
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Homotopy types and acyclic relations
Say $X$, $Y$ are two compact Hausdorff topological spaces - in my mind, one can take $X, Y$ to be both included in $\mathbb{R}^d$ - of the same homotopy type, i.e there exist $f : X \to Y, g: Y \to X$ ...
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Is there a general theory of CW-approximation?
Suppose $\mathcal C$ is a model category or an infinity category. For simplicity I will assume that $\mathcal C$ is stable, but this may not be essential. I am interested in a theory of CW ...
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Opposite of cyclification
I'm learning about the idea of cyclification in rational homotopy theory, and it encodes double dimensional reduction in physics. I was wondering whether there is any idea to "decompactify" ...
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Monad interleaving
In Eugenia Cheng's paper Monad interleaving: a construction of the operad for Leinster's weak $\omega$-categories the adjoint functor for the forgetful functor $\mathbf{OWC} \to \mathbf{Coll}$ is ...
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Warner's elegant papers: conjugate locus, sprays, and fundamental groups
Motivation
I am working on the problem of uniqueness of Frechet mean for probability measures on a Riemannian manifold of non-negative curvature, for which I need to understand properties and ...
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Does the "nerve functor" from cubical sets to simplicial sets preserve weak equivalences?
Denote by $\Box$ the box category with connections, and define the category of cubical sets (with connections) ${\sf cSet}$ as the category of presheaves over $\Box$. Consider the adjunction
$$
{\sf ...
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Injective model structure on $\mathrm{Ch}^{\geqslant 0}(R)$
$\DeclareMathOperator\Ch{Ch}$Let $\Ch^{\geqslant 0}(R)$ be a category of non-negatively graded cochain complexes over a ring $R$.
It is a general knowledge that if we consider the following three ...
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Generating the $\infty$-category of $\infty$-categories using $\infty$-presheaf topos
Let $D$ be an $\infty$-category. I’m curious when $\mathcal{Cat}_\infty$ is an accessible reflective localization of the $\infty$-presheaf topos $\mathrm{PShv}(D)$.
This reduces to study the property ...
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Derived formulation of étale Poincaré duality
I’m looking for some answers on the appropriate general formulation of Poincaré duality in étale cohomology. Excuse my highfalutin formulation, my background is more in homotopy theory than in ...
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Convergence of Ext Spectral Sequence for modules over $\mathbb{E}_1$-ring spectra
Let $R$ be an $\mathbb{E}_1$-ring spectrum and $M,N$ right $R$-module spectra. In Variant 7.2.1.24 in Lurie's Higher Algebra, he remarks the existence of a spectral sequence $E_r^{p,q}$ with $E_2^{p,q}...
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Completion of cohomology of derived completion
I am by far not an expert in this area. I've tried many approaches, but I always run into some technicality.
So, let $R$ be some commutative ring, $u$ a degree two variable, and consider $R[u]$ as a ...
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Non-nilpotent elements in the cohomology of the mod-2 Steenrod algebra
I have heard that there are interesting non-nilpotent elements on the $E_2$-page of the mod-2 Adams spectral sequence (besides just $h_0$). For example, I have heard that $g$ and some related elements ...
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Ordered Čech nerve
Given a topological space $X$, an abelian sheaf $\mathcal{F}$, and a covering $X = \bigcup_{i \in I}U_i$, we can compute the sheaf cohomology using the Čech complex, whose $k$-th term is given by $$\...
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