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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
Is the motivic homotopy spectrum of Hermitian K-theory $\eta$-complete?
If I understand the question correctly, you're asking if there are base fields $k$ of characteristic not $2$ and such that $k[\sqrt{-1}]$ has a finite $\mathbb{Z}/2$-cohomological dimension (the assum …
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4
votes
Accepted
Compact objects in the $\infty$-category presented by a simplicial model category
If $X$ is such that $X \times \Delta^n$ is compact for every $n$ then yes. This happens, for example, if the cotensor functor $(-)^{\Delta^n}$ preserves filtered colimits, a condition which is quite c …
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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
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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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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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
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
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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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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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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1
vote
Construction for algebras over little cubes operad
As you point out, it indeed seems that in order to get a well-behaved answer one should work in a suitable homotopical setting, for example, that of $\infty$-categories and $\infty$-operads. Using thi …
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5
votes
Accepted
Criterion for homotopy pullback square of simplicial categories
Yes. In fact such a square can be replaced with a weakly equivalent Reedy fibrant pullback square without changing the object set of any of the simplicial categories. For a proof see, e.g., Lemma 3.1. …
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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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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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4
votes
Accepted
Homotopy theoretic description of homotopy fixed points (and obstructions) for an action of ...
I'm not sure exactly what kind of answer you're looking for, but I can try to give some context which may make things sound more reasonable. Let us think of of groupoids as $1$-truncated $\infty$-grou …
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