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Results tagged with homotopy-theory
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user 51164
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.
60
votes
Accepted
What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
Here is my guess. To compare spaces with their notion of strict $\infty$-groupoids (in which everything is strict except inverses) Kapranov and Voevodsky use an intermediate category of Kan diagrammat …
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26
votes
Accepted
Why study the p-completions of a space?
First one should separate between the property and being $p$-complete and process of $p$-completion. In the classical setting, the $p$-completion functor is not so well-behaved for general spaces. For …
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15
votes
Why Grothendieck's Homotopy Hypothesis is so difficult?
First, let me remark that your question does not seem to be about the homotopy hypothesis, but about rectification. More specifically, the homotopy hypothesis concerns the question of whether (some ve …
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14
votes
Sheaves of complexes and complexes of sheaves
If $A$ is a Grothendieck abelian category then $Sh(X,A)$ is a Grothendieck abelian category, in which case one can endow the category $C(Sh(X,A))$ of unbounded complexes in $Sh(X,A)$ with the injectiv …
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13
votes
1
answer
474
views
Is the operadic nerve functor an equivalence of ∞-categories?
It is now known that the $\infty$-category of $\infty$-operads as defined by Lurie is equivalent to the underlying $\infty$-category of the model category of simplicial operads, see http://arxiv.org/p …
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9
votes
Accepted
The cofibration/fibration $\leftrightarrow$ epi/mono confusion
The (epi,mono) factorization system in Sets is part of a model structure on Sets whose weak equivalences are the epis, fibrations are monos and cofibrations are everything. This is a model for the hom …
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9
votes
Accepted
Is every locally compactly generated space compactly generated?
The paper "A distinguishing example in k-spaces" by John Isbell constructs an example of a locally compact space $X$ which is not compact-Hausdorffly generated.
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8
votes
Accepted
Quillen equivalence, fibrant objects
Here is a counter-example to the dual assertion (so that you can get a counter-example to your original question by taking the opposite model categories). Consider the category ${\rm Set_\Delta}$ of s …
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7
votes
From relative categories to marked simplicial sets
Concerning the first question: the simplicial localization functor $L^H$ induces an equivalence from the relative category of small relative categories to the relative category of small simplicial cat …
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7
votes
Accepted
About fibrations with fibre Eilenberg-MacLane spaces
No. If this were the case then there would be a section $s: B \to E$ to $f$ induced by the $G$-equivariant map $\widetilde{s}:\widetilde{B} \to \widetilde{B} \times {\rm K}(M,n)$ sending $x$ to $(x,0) …
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7
votes
Accepted
Methods for defining/calculating homotopy limits of quasicategories
When working with quasi-categories, it is often more convenient (and more compatible with existing machinery) not to work with actual strict diagrams of quasi-categories but rather with coCartesian fi …
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6
votes
Accepted
Spelling out explicitly the data of a two step filtration in terms of pieces and gluing data
Technically speaking the answer to your question is no, in the sense that the data of $(\alpha,\beta,\gamma,\delta)$ alone does not determine the filtered object $V_0 \subseteq V_1 \subseteq V_2$. How …
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6
votes
Accepted
Waldhausen $K$-theory before group completion
I'm not sure about Waldhausen categories in general, but if you restrict attention to stable $\infty$-categories (with trivial Waldhausen structure in which all maps are cofibrations) then group compl …
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6
votes
Accepted
How is topological André-Quillen homology (TAQ) a "stabilization", exactly?
These stabilization formulas do indeed follow from the paper of Basterra-Mandell. Fix a commutative $S$-algebra $A$. Then Basterra and Mandell prove the following:
1) [Theorem 3] Given a commutative …
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6
votes
Accepted
Compatibility of Grothendieck construction with pullback
Yes, though it is usually written as the commutativity of unstraightening with pullback (on the $\infty$-categorical level it doesn't matter, since straightening and unstraightening are inverse equiva …
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