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Results tagged with homotopy-theory
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user 18263
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.
10
votes
How much of homotopy theory can be done using only finite topological spaces?
Peter May has been working on an entire book (or maybe just a comprehensive set of lecture notes?) addressing your exact question (and much more). The preprint version which he shared with me is calle …
- 15.5k
15
votes
What is the intuitive meaning of the coskeleton of a simplicial set?
A simplicial set $X$ is $k$-coskeletal iff the following condition holds:
a simplex of dimension $\geq k$ is present iff all of its $(k-1)$-dimensional faces are present in $X$.
A standard exa …
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5
votes
0
answers
152
views
Contractibility of a poset-indexed colimit
Let $(X,\leq)$ be a poset with distinguished element $p$, and let $P'$ be the poset of "finite chains which weakly descend to $p$" given by all $\sigma = (x_0 \geq x_1 \geq \cdots \geq x_k \geq p)$ or …
- 15.5k
4
votes
Is there an analog of Sperner's lemma for the Hopf invariant?
It seems really hard to impose combinatorial Sperner-like conditions which would guarantee the nontriviality of the Hopf invariant. But if you allow things to get slightly more algebraic by constructi …
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3
votes
2
answers
201
views
When is the Morse equivalence local?
Let $f:X \to \mathbb{R}$ be a Morse function on some compact submanifold $X \subset \mathbb{R}^n$, and assume that $p \in X$ is not a critical point of $f$. For some $\epsilon > 0$ let $D_\epsilon(p)$ …
- 15.5k
3
votes
When are maps between topological spaces homotopic?
Mark Grant's answer provides a good class of examples, but a slightly more general class can be found after (many, many hours) of reading Whitehead's "Combinatorial Homotopy II" available here.
I th …
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13
votes
1
answer
469
views
When does localization preserve homotopy type of classifying spaces?
Let $\mathcal{C}$ be a small category and $\Sigma$ a collection of morphisms in $\mathcal{C}$. Denote by $F_\Sigma:\mathcal{C} \to \mathcal{C}[\Sigma^{-1}]$ the usual quotient functor from $\mathcal{C …
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21
votes
2
answers
1k
views
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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7
votes
Discrete Morse theory: how do zig-zag paths determine homotopy type?
Thanks to Cosheaf Overlord Justin Curry for bringing this question to my attention. I'm only going to address the first question here, and I think with some computations (whose complexity depends on y …
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28
votes
1
answer
1k
views
Is there a general theory of fiber theorems?
Here are three vague theorems rolled up in one.
Let $X$ and $Y$ be sufficiently nice topological spaces and $f:X \to Y$ a sufficiently nice surjection. If for each $y \in Y$, the fiber $f^{-1}(y) …
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3
votes
0
answers
305
views
Are there CW structures on homotopy limits of CW maps?
Consider the diagram of finite CW complexes $X \stackrel{f}\leftarrow Y$ where $f$ is a cellular map and note that its homotopy colimit is precisely the mapping cylinder
$$C_H = \frac{X \sqcup (Y \tim …
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5
votes
2
answers
436
views
How does one Segal-subdivide a 2-category?
Let $\mathcal{C}$ be a small category. Then, its Segal subdivision $\text{sd }\mathcal{C}$ is a new category whose objects are morphisms of $\mathcal{C}$, and a morphism from $f:x \to y$ to $g: w \to …
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5
votes
0
answers
230
views
Adding morphisms to a category without changing homotopy type
I have a really tame category $C$: there are only finitely many objects $C_0$, each hom-set $C(x,y)$ for $x,y \in C_0$ has at most one element, and (aside from identity morphisms) if $C(x,y)$ is non-e …
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3
votes
2
answers
666
views
Zigzags and contractibility of categories
Let $\mathbf{C}$ be a small category and $\mathbf{C}'$ its hammock localization in the sense of Dwyer and Kan. I am looking for a proof (or counterexample) of the following assertion:
If there is …
- 15.5k
7
votes
Accepted
Homotopy theory of acyclic categories
Here is a cool new (and very readable) preprint which uses the second barycentric subdivision (as discussed in Zhen Lin, Fernando Muro and Peter May's comments) to construct a cofibrantly generated mo …
- 15.5k