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Results tagged with homotopy-theory
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user 12547
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.
5
votes
Accepted
homotopy tensor product of functors and bar construction
Yes. (Of course for these constructions to be homotopically well-behaved you need $F$ to be levelwise cofibrant.) In fact, assuming that such $QK$ exists we can express it by an explicit formula which …
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2
votes
Cube Lemma on a cofibrantly generated (almost) model category
This is not an answer in a full generality, but it is certainly not the case if the domains of generating acyclic cofibrations are cofibrant. (I have a hard time thinking of an example of a model cate …
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5
votes
Accepted
Does the right adjoint of the category of simplices functor is "homotopicaly inverse" to the...
This indeed true and is discussed in depth by Latch, Thomason and Wilson in a paper called Simplicial Sets from Categories. Your question is answered in Corollary 4.7 and relies on Theorem 4.1, the ma …
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6
votes
Accepted
Freely adding degeneracies does not change the homotopy type
Here is a purely combinatorial and straightforward proof.
First, you can easily check that $\Delta[m]'$ is the nerve of the category $[m]'$ that is obtained from the poset $[m]$ by freely adjoining o …
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14
votes
Accepted
When are (weak) homotopy equivalence testable on open covers?
For weak homotopy equivalences this holds always (Theorem 6.7.9 in tom Dieck's Algebraic Topology).
For homotopy equivalences this holds provided the open covers are numerable (Theorem 4.2.7 loc. cit …
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8
votes
Accepted
Homotopy pullbacks/relative homotopy groups vs homotopy pushouts/relative homology groups
Relative homotopy groups are the homotopy groups of the homotopy fiber. A homotopy pullback square induces an equivalence of the homotopy fibers of two of its parellel maps by the cancellation propert …
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5
votes
Suspension of an excisive pair
The answer to your first question is negative. Before I give a counterexample, let me rephrase the problem in terms I consider more natural.
First, I believe it is more convenient to consider excisiv …
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6
votes
Alternative characterization of homotopy equivalence
EDIT: This is an old question, but I have stumbled upon it by accident and realized that my answer is wrong. It turns out that genuine homotopy equivalences cannot be characterized in terms of their h …
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6
votes
Accepted
mapping spaces of diagrams
Let me mention yet another approach using the simplicial localization due to Dwyer and Kan.
Given any category $\mathcal{C}$ equipped with a class of morphisms $W$ one can form a simplicial category …
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7
votes
Accepted
Sheaf cohomology invariant of weak homotopy type?
No. For paracompact spaces sheaf cohomology coincides with Čech cohomology. In particular it applies to the closed topologist's sine curve $C$. There is a map $C \to S^1$ inducing an isomorphism on Če …
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5
votes
General gluing theorem for adjunction spaces
Yes, this is true even for not necessarily closed cofibrations. If you want a single source that gives a complete proof, then the only one that comes to my mind is this preprint.
Definition 1.1.1 int …
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7
votes
Accepted
Do finite simplicial sets jointly detect isomorphisms in the homotopy category?
The answer is no. Otherwise, it would follow from Brown's representability theorem (and here I mean very specifically Theorem 2.8 from Brown's 1965 paper Abstract Homotopy Theory) that every "half-exa …
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4
votes
Maps with Hopf invariant zero are suspensions
There is a $2$-local fiber sequence
$$S^n \to \Omega S^{n+1} \to \Omega S^{2n+1}$$
where the first map is the suspension map. Its associated long exact sequence of homotopy groups is called the EHP …
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15
votes
Accepted
Can homotopy pullbacks of spaces be checked on fibers?
$\require{AMScd}$I don't know a reference but the proof is easy enough. Form homotopy pullback squares
\begin{CD}
Fu @>>> Ff @>>> A \\
@VVV @VVV @V{u}VV \\
* @>>> Fg @>>> P @>>> B \\
@. @VVV @VVV @VV …
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6
votes
Accepted
Simple characterization of Postnikov & Whitehead towers?
Indeed, the properties you stated characterize Postnikov and Whitehead towers. A nice conceptual way of justifying this is by using $k$-connected / $k-$truncated factorization systems.
To fix termino …
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