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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.
3
votes
Accepted
The locally model bicategory of $\cal V$-profunctors
Given two model categories $\mathcal{M},\mathcal{N}$, one does know what would have been a left Quillen functor out of what would have been the tensor product $\mathcal{M} \otimes \mathcal{N}$ into a …
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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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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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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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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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5
votes
Accepted
real and complex vector spaces as topological categories
I think the answer is no. Suppose there exists an enrichment satisfying your requirements, and let $U: Vect_{\mathbb{C}} \to Vect_{\mathbb{R}}$ be the forgetful functor. Let $C \subseteq Map(\mathbb{R …
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5
votes
Accepted
Proposition in HTT on cofibrations of categories
You can argue as follows. Suppose that $g: D \to D'$ is a retract of $f: C \to C'$ (in the category of $S$-enriched categories) via maps $D \stackrel{i}{\to} C \stackrel{r}{\to} D$ and $D' \stackrel{i …
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5
votes
Accepted
A finite Whitehead Theorem for $\infty$-topos
Let $\mathcal{X}$ be the $\infty$-topos in question containing an object $X \in \mathcal{X}$. I assume that by $X$ having homotopy dimension $\leq n$ you mean that the $\infty$-topos $\mathcal{X}_{/X} …
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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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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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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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5
votes
Property-like structure in a model category
Specifically for the case of quasi-categories (or any other model for $\infty$-categories) the following observation can be useful: suppose that $f: {\cal C} \to {\cal D}$ is a map of quasi-categories …
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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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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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2
votes
Accepted
Morphisms of parametrized ring spectra
So the answer is a bit surprising (maybe I have a mistake). You have an adjunction between $Fun(X,\mathrm{Sp})$ and $\mathrm{Sp}$ which in one direction sends a functor to its (homotopy) colimit and o …
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