All Questions
Tagged with localization homotopy-theory
15 questions
8
votes
1
answer
440
views
Model categories as a tool to resolve size issues for localizing categories
I have a rather basic question about one motivation for introducing model categories in David White's notes, as a possible way to overcome troubles with size issues appearing when localizing a ...
- 30.3k
6
votes
2
answers
390
views
Size issues in localization $\mathcal{C}[\mathcal{W}^{-1}]$ category
When one starts with a locally small category $\mathcal{C}$ and wants to localize it at an appropriate choosen collection of morphisms $\mathcal{W}$, then in general one faces some size issues in the ...
- 10.1k
7
votes
1
answer
303
views
Localisation of categories, but instead of "isomorphisms" I want "morphisms with right inverse". Construction via calculus of fractions possible?
Let $\mathcal{C}$ be a category and $S$ a collection of morphisms in $\mathcal{C}$. The localisation $\mathcal{C}[S^{-1}]$ is defined via a functor $F: \mathcal{C} \longrightarrow \mathcal{C}[S^{-1}]$ ...
- 30.3k
5
votes
1
answer
174
views
Slices for certain $C_p$-spectrum
By the work of Hill-Yarnall, for the group $G=C_p,$ all the slices for any spectrum, in particular, for $S^V \wedge H\underline{\mathbb{Z}}$, are classified. Here $V$ is a representation of $C_p.$
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- 13.5k
22
votes
1
answer
679
views
When does rationalization commute with homotopy fixed points?
Let $X$ be a $G$-space. There are a number of places in the literature where one can find the claim that under certain conditions rationalization and taking homotopy fixed points with respect to a ...
- 8,167
4
votes
0
answers
310
views
A question related to bousfield localization and nilpotent completion
I am reading Bousfield's paper entitled "The localization of spectra with respect to homology" (MSN). In that paper, Corollary 6.13 states that, if a ring spectrum $E$ has countable homotopy and ...
- 745
5
votes
0
answers
138
views
Analog of cellular approximation theorem for $CW_0$-complexes ($CW_\mathcal P$-complexes)
$CW_0$-complexes are analogs of $CW$-complexes, in which the "building blocks" are the rational disks $D^{n+1}_0$ whose boundaries are given by $\partial D^{n+1}_0= S^n_0$, where $S^n_0$ is a ...
- 523
19
votes
1
answer
958
views
What is classified by generalised Eilenberg MacLane spaces?
Given an abelian group $A$, the Eilenberg MacLane spaces $K(A,n)$ represent the the nth cohomology group in $A$.
In a similar vein, given an arbitrary group $G$ and a space $X$, maps to the ...
- 35.9k
30
votes
1
answer
787
views
Is a filtered colimit of rational spaces again rational?
Let me first explain the statement of the question and then give some indication why the answer might be 'yes'.
By a space I mean, say, a simplicial set and by rational I mean rational in the sense of ...
- 52.7k
6
votes
1
answer
321
views
Localization, Slice Tower, and Motivic Spectra
Suppose $k$ is an algebraically closed field of characteristic $p>0$. There is an $\infty$-category of motivic spectra over $k$, denoted $\mathcal{S}pt(k)$. As in algebraic topology, there are ...
- 16.5k
10
votes
1
answer
763
views
Example of a saturated class of morphisms which is not _obviously_ saturated?
By "saturated class of morphisms" in a category $\mathcal{C}$, I mean a subcategory $\mathcal{W} \subset \mathcal{C}$ such that the image of $\mathcal{W}$ in $\mathcal{C}[\mathcal{W}^{-1}]$ consists ...
- 66.8k
5
votes
0
answers
225
views
Weak equivalences of left Bousfield localizations
Suppose C is a complete and cocomplete category with two model structures (C0,F0,W0) and (C1,F1,W1) such that C0⊃C1, F0⊂F1, W0⊂W1.
If necessary, the model structures can be assumed to be simplicial, ...
- 37.8k
3
votes
1
answer
433
views
Bousfield localization before and after taking homotopy
The ncatlab says:
Under suitable conditions it should be true that for $C$ a model category whose homotopy category $\mathrm{Ho}(C)$ is a triangulated category the homotopy category of a left ...
- 15.2k
9
votes
1
answer
393
views
Homotopical descent information contained in the Dwyer-Kan function complexes of a presheaf category?
Recall that the category of sheaves on some site $C$ equipped with a grothendieck topology $\tau$ is equivalent to the localization of the category of presheaves $W^{-1}Psh(C)$ at $W$ where $W$ is the ...
- 66.8k
3
votes
1
answer
317
views
Controlling Reflective Subcategories and Localizations
Localizations are an extremely important part of modern homotopy theory. Both the category of spaces an spectra have a plethora of interesting localizations: at a fixed prime, rational, with respect ...
- 3,622