All Questions
17
questions
5
votes
0
answers
356
views
Why do homotopy orbits commute with the Eilenberg–Mac Lane spectrum functor?
Let $H: \mathsf{sAb}\to H\mathbb{Z}\text{-}\mathsf{Mod}$ denote the Eilenberg–Mac Lane functor sending a simplicial abelian group $M_\bullet$ to the infinite delooping of its geometric realization, i....
3
votes
1
answer
163
views
Which positive flat stable model structures on (flavors of) spectra have the property that cofibrant operad-algebras forget to cofibrant spectra?
Let $M$ be a monoidal model category and $O$ an operad valued in $M$, and the category of $O$-algebras inherits a model structure from $M$ where a map $f$ is a weak equivalence (resp. fibration) if ...
6
votes
3
answers
686
views
Why does this construction give a (homotopy-invariant) suspension (resp. homotopy cofiber) in an arbitrary pointed model category?
In their text Foundations of Stable Homotopy Theory, Barnes and Roitzheim define the suspension of a cofibrant object X of a pointed model category to be the pushout of the diagram $*\leftarrow X\...
- 1,837
7
votes
1
answer
922
views
Homotopy pullbacks and pushouts in stable model categories
There are lots of similar questions that have been answered on this topic (particularly Homotopy limit-colimit diagrams in stable model categories), but I have a specific question that I do not ...
user142851
11
votes
0
answers
324
views
$\Gamma$-sets vs $\Gamma$-spaces
I know that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-space, i.e. a $\Gamma$-set.
For example, Pirashvili proves, as theorem 1.2 of Dold-Kan Type Theorem for $\Gamma$-Groups, ...
- 40.8k
9
votes
1
answer
510
views
Are cofibrant commutative S-algebras flat?
Let $R$ be a cofibrant commutative $S$-algebra (in the sense of Elmendorf-Kriz-Mandell-May; they call them "$q$-cofibrant") and $A$ be a cofibrant commutative $R$-algebra.
Does $A\wedge_R-:RMod→...
- 2,944
5
votes
1
answer
470
views
Stable Dold-Kan correspondence and symmetric group actions
There exists a Quillen equivalence between $HRModSpectra$ (model category of ring spectra over Eilenberg-MacLane spectra $EM(R)$, where $R$ is a commutative ring, with stable model structure) and $Ch$ ...
- 371
63
votes
5
answers
17k
views
What is modern algebraic topology(homotopy theory) about?
At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other ...
13
votes
1
answer
1k
views
Fibrant-cofibrant models of Eilenberg-MacLane spectra
There are many models for spectra, by which I mean a model category whose homotopy category is triangulated-equivalent to the stable homotopy category. In each model, there are ways to construct ...
- 1,223
3
votes
1
answer
279
views
Triangulated structure on $\mathbf{SH}(S)$: $\mathbb{P}^1$-suspension versus classical suspension
I am studying the construction of the motivic stable homotopy category of schemes $\mathbf{SH}(S)$ following Riou's paper Categorie homotopiquement stable d'un site suspendu avec intervalle (click to ...
- 2,811
6
votes
1
answer
602
views
Stabilization of a generic pointed model category
Let $\mathcal C$ be a pointed model category. It is well-known that its homotopy category $\mathrm{Ho}(\mathcal C)$ is naturally a $\mathrm{Ho}(\underline{\mathrm{sSet}}_*)$-category, where $\mathrm{...
- 2,399
31
votes
4
answers
4k
views
Fibrations and Cofibrations of spectra are "the same"
My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is:
"For spectra every cofibration is equivalent to a fibration" (e.g. in the ...
- 1,538
5
votes
2
answers
705
views
On triangulated categories of pro-objects
Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?
I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) ...
- 16.7k
5
votes
2
answers
2k
views
Homotopy limit-colimit diagrams in stable model categories
It is shown in Remark 7.1.12 of (a newer version of) Mark Hovey's book Model Categories that, in a stable model category, homotopy pullback squares coincide with homotopy pushout squares. The argument ...
- 3,154
8
votes
1
answer
433
views
Monoidal Model Categories with Suspension Functor
This is basically just me trying to find out what such categories are called, and where they are written about. If I think of some model category of spectra being a "stabilization" of some model ...
- 10.3k