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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.
19
votes
2
answers
2k
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Does the bordism homology theory satisfy the weak equivalence axiom?
There is an interesting and important homology theory called bordism. Briefly speaking, a singular manifold in a space $X$ is a pair $(M, f)$ where $M$ is a closed smooth manifold and $f : M \to X$ is …
- 7,656
16
votes
Accepted
On a weaker version of homotopy equivalence between topological spaces
Saying that there are maps $f \colon X \to Y$ and $g \colon Y \to X$ such that $g f$ is homotopic to $\mathrm{id}_X$ means that $X$ is a homotopy retract of $Y$. (By the way, we say that maps are homo …
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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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15
votes
Accepted
Constructing a "geometric" model structure on Cat by localizing the "categorical" model stru...
No, there are far too many "canonical cofibrations" for that. For example, let $A$ be a category with two objects and two parallel arrows between them. Then there is a unique functor $A \to [1]$ that …
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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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12
votes
Accepted
When do colimits agree with homotopy colimits?
I don't think we can expect to have one general answer to this question, only a collection of unrelated specialized results. Here are two more:
In the category of simplicial sets all filtered colimi …
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10
votes
Accepted
Can a weak fibration category be non saturated?
It is a result of Cisinski that in a fibration category the three conditions you mention (saturation, 2-out-of-6, weak equivalences closed under retracts) are all equivalent. See Theorem 7.2.7 in this …
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10
votes
Accepted
Relative version of Quillen's theorem A
The condition that appears in the assumption of what you call "Relative Theorem A" was introduced by Grothendieck in Pursuing Stacks. It is a part of the definition of a basic localizer, i.e. a class …
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9
votes
Accepted
When does localization preserve homotopy type of classifying spaces?
You can find some sufficient conditions in terms of simplicial localization of Dwyer and Kan. In Prop. 3.7 of Simplicial Localizations of Categories they prove that it holds when $\Sigma$ is free and …
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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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8
votes
1
answer
526
views
Localizations of non-nilpotent spaces
For simplicity let's talk about $p$-localizations of spaces for a fixed prime $p$. Every space $X$ has a well-defined $p$-localization which can be constructed by the small object argument and which b …
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7
votes
Accepted
When does simplicial localization commute with functor categories?
For general $C$ (I will drop $W$ from the notation) this is not true even when $D$ is an infinite discrete category.
In the answer to this similar question I described how to construct a sequence of …
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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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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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7
votes
Can one make the category of pairs of topological spaces a model category?
As Tyler Lawson points out you can use the category of all diagrams on $[1]$. Then the projective and injective model structures are both instances of Reedy model structures. This is discussed in Sect …
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