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Results tagged with homotopy-theory
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user 11546
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
How does homotopy theory simplify topology but allow for complexity in higher category theory?
I think one way to look at this is to say that, from a certain point of view, topological spaces are far more complicated than categories and homotopy types are somewhere in the middle. The only wrink …
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4
votes
1
answer
152
views
Are monomorphisms in an $\infty$-topos preserved by $0$-truncation?
Let $\mathfrak{X}$ be an $\infty$-topos and let $f\colon X\to Y$ be a morphism of $\mathfrak{X}$. We say that $f$ is a monomorphism if it is $(-1)$-truncated which means that for every $Z\in\mathfrak{ …
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7
votes
0
answers
173
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Simplicial right Kan extensions and Cartesian transformations
I will write the concrete question first, in case the answer is clear independently of the context:
Question: Given an $\infty$-topos $\mathfrak{X}$ and a diagram $F\colon\Delta^1\times\Delta_+^{op}\ …
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4
votes
Accepted
The first two $k$-invariants of $\mathrm{pic}(KU)$ and $\mathrm{pic}(KO)$
Essentially building on Chris Schommer-Pries' comment above, this has been worked out by Kiran Luecke, Jack Morava and myself in Section 4.2 of https://arxiv.org/pdf/2306.10112.
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5
votes
2
answers
370
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Monomorphisms of diagrams in an $\infty$-category
Let $f,g\colon K\to \mathcal{C}$ be diagrams in a nice $\infty$-category $\mathcal{C}$. I have two general questions:
If I have a natural transformation $\eta\colon f\Rightarrow g$ which is a monomor …
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2
votes
Accepted
Quasicategorical Construction of a Cosimplicial Map of Rognes
This question has been answered by the PhD thesis of Aras Ergus. See Corollary 3.2.8 here: https://infoscience.epfl.ch/record/295824/files/EPFL_TH9067.pdf.
The basic idea is to recognize that the como …
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9
votes
0
answers
566
views
The relation between the motivic Galois group and the motivic Steenrod algebra
There is a point of view on the Steenrod algebra that goes something like the following: the functor $-\otimes H\mathbb{F}_p\colon Mod_{\mathbb{S}}\to Mod_{H\mathbb{F}_p}$ corresponds to pulling back …
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5
votes
0
answers
157
views
Splitting of $BGL_1(KR)$
There are infinite loop space splittings $BGL_1(KO)\simeq BGL_1(KO)[0,2]\times Z$ and $BGL_1(KU)\simeq BGL_1(KU)[0,3]\times Z'$ where $Z$ and $Z'$ are 2 and 3 connected, respectively (i.e. they have t …
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14
votes
1
answer
348
views
The first two $k$-invariants of $\mathrm{pic}(KU)$ and $\mathrm{pic}(KO)$
$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\pic{pic}$Real and complex topological $K$-theories, $KO$ and $KU$, have Picard spectra $\pic(KO)$ and $\pic(KU)$ built from the $\mathbb{E}_\infty$-s …
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1
vote
Why does $Mf$ always support an $Mf$-orientation?
I just want to add another answer to this, which is Corollary 4.16 of arXiv:1810:00734. This result of course uses Omar and Tobias' in an essential way, so is not somehow independent, but what it does …
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5
votes
0
answers
173
views
(Co)homology of a directed space with coefficients in a commutative monoid
This is essentially a reference request, or a request for an explanation of why this cannot be done in a useful or interesting way (i.e. an explanation of why no such reference exists!).
If I have a d …
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4
votes
Accepted
Lifting Strict Comonoids and Comodules to Quasicategories
The answer to this question is yes, and it's the main result of this paper.
One thing to point out is that, even in the case that the tensor product of $\mathcal{M}$ preserves fibrant objects (so tha …
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4
votes
Accepted
Interaction of Grothendieck Construction with Coherent Nerve
Answering this question took me some time. First of all, Liang Ze Wong and I had to write down a version of the enriched Grothendieck construction that worked for these purposes, as Tamaki's construct …
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9
votes
1
answer
289
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Essential maps of spectra which are null when localized at any prime
There are maps of spaces which are not null-homotopic, but when localized at any prime become null. I don't know explicit constructions of any, but an example is given in Section 6 of Chapter 25 of th …
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15
votes
2
answers
920
views
What homotopy classes can attaching an $E_n$-cell kill?
Let $A$ be a connected $E_{n+1}$-ring spectrum and let $\alpha\in\pi_k(A)$. I am having trouble showing that attaching an $E_n$-cell along $\alpha$ will necessarily not kill an element $\beta\in\pi_k( …
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