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Results tagged with homotopy-theory
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user 2503
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.
37
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Conjectures in Grothendieck's "Pursuing stacks"
I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to this questi …
- 35.5k
17
votes
Why is the motivic category defined over the site of smooth schemes only?
It makes sense to consider larger versions of the (unstable and stable) motivic homotopy categories built out of the site $Sch_S$ of all schemes over $S$ (say of finite type to avoid dealing with size …
- 5,901
24
votes
Why do we need model categories?
This answer is an elaboration on Dylan's comments.
1) Let us define a homotopy theory to be a pair $(C, W)$, where $C$ is a category and $W$ is some class of morphisms called weak equivalences.
(Let' …
- 5,901
14
votes
What is the relationship between connective and nonconnective derived algebraic geometry?
As Tyler pointed out, it is "too easy" to be representable in the non-connective world. This might sound good, but it comes at the cost of geometric intuition. It is related to the fact that negativ …
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7
votes
Morphisms of $\mathbb E_l$-rings between $\mathbb E_k$-rings for $l<k$
Let $R$ be an $E_\infty$-ring spectrum and write $R\{t\}$ (resp. $R[t]$) for the free $E_\infty$-$R$-algebra (resp. free $E_1$-$R$-algebra) on one generator $t$ (in degree zero). This notation is com …
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7
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Accepted
When do the polynomial algebra and free algebra coincide in brave new algebra?
Any morphism of $R$-algebras $\varphi : R\{t\} \to R[t]$ is determined up to homotopy by an element of $\pi_0(R[t]) \approx \pi_0(R)[t]$. If $\varphi$ is an equivalence, then this element must be the …
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23
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Accepted
Motivation and potential applications of spectral algebraic geometry
This is not really an answer to your question, just an attempt to address your question from the comments.
There are various flavours of homotopical or higher algebraic geometry that are commonly con …
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9
votes
How to show the following two definitions of homotopy monomorphism are equivalent?
Let $sSet$ be the category of simplicial sets with the Quillen model structure. Define a homotopy monomorphism in $sSet$ to be a morphism whose homotopy fibres are empty or weakly contractible. In a …
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3
votes
A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categori...
In some sense, the "universal" version of this fact was proved by Blumberg-Gepner-Tabuada as Proposition 3.3 in this paper.
That is, they proved the analogue for stable $\infty$-categories, which is t …
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3
votes
Accepted
Descent properties of spaces
For the first problem, as I wrote in the comments, the author is implicitly using the language of $(\infty,1)$-categories, where it does make sense to speak of such a functor.
To understand why, you c …
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25
votes
Algebraic K-theory and Homotopy Sheaves
The question already has good answers but I think there is still more to be said.
References
As already mentioned, algebraic K-theory satisfies Zariski descent. For regular noetherian schemes this …
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5
votes
When are homotopy categories of model categories closed modules over the homotopy category o...
I am not sure if this will answer your question, but it may at least point you in the right direction (or at least some direction).
Let me start with some classical background.
Let $C$ be a category …
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