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Results tagged with homotopy-theory
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user 24563
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
1
answer
323
views
About (co)limits of accessible categories
I am reading the paper colimits of accessible categories. In the introduction, the authors summarize what is known about limits and colimits of accessible categories. I believed that there was somethi …
- 3,901
6
votes
3
answers
525
views
Transporting a model category structure along a left adjoint
There is a well-known theorem for transporting a model category structure along a left adjoint $F:\mathcal{M}\to \mathcal{N}$ which is explained here and which is due to Sjoerd Crans.
The difficult p …
- 3,901
6
votes
0
answers
232
views
About a zig-zag of Quillen adjunctions
I have the following situation:
Three combinatorial model categories $\mathcal{K}_1, \mathcal{K}_2, \mathcal{K}_3$ such that all objects are fibrant (so no need to bother with the fibrant replaceme …
- 3,901
3
votes
0
answers
331
views
About the Moore composition of paths
1) QUESTION (EDIT: 04/28/2020 to remove a possible counterexample)
I work with weak Hausdorff $k$-spaces (so all spaces are $T_1$). The internal hom is denoted by $\mathbf{TOP}(-,-)$. Let $\mathcal{G …
- 3,901
3
votes
0
answers
167
views
Calculation of the homotopy colimit of a diagram of spaces
Consider a small category $I$. There exists a small diagram $D:I^{op}\to {\rm Top}$ where ${\rm Top}$ is a convenient category for doing algebraic topology such that for all small diagrams $X:I\to {\r …
- 3,901
2
votes
Accepted
Transporting a model category structure along a left adjoint
It suffices to dualize the proof of Theorem 2.2.1 in Necessary and sufficient conditions for induced model structures. It uses indeed an argument coming from Quillen's book "Homotopical Algebra", II p …
1
vote
0
answers
138
views
Identifying the two points of a subspace homeomorphic to a Sierpinski space
Let $X$ be a $\Delta$-generated space having a subset $A=\{a,b\}$ such that the relative topology is the Sierpinski topology with for example $\{a\}$ closed and $\{b\}$ open (the Sierpinsky space is a …
- 3,901
5
votes
1
answer
173
views
Fibrant replacement of an injective model category of enriched diagrams
Take a topologically enriched small category $\mathcal{P}$ and the category of enriched diagrams of spaces $[\mathcal{P},\mathrm{Top}]_0$. We work with the category of $\Delta$-generated spaces equipp …
- 3,901
3
votes
0
answers
98
views
About homotopy weighted colimit
Let $X:I\to M$ be a functor from a small category $I$ to a simplicial accessible (not combinatorial) proper model category $M$ which is already injective cofibrant. Let $W:I^{op}\to \mathrm{Set}$ be a …
- 3,901
8
votes
0
answers
367
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Model category of cofibrant topological spaces
By browsing the list of open problems in Mark Hovey's book "Model categories", I came across the following one (Problem 8.3): find a model structure on topological spaces, with the same weak equivalen …
- 3,901
8
votes
0
answers
217
views
Cubical model category
Question. Call a cubical model category a model category enriched over cubical sets equipped with a tensor product $X \otimes K$ and a cotensor product $X^K$ where $X$ is an object of the model categ …
- 3,901
9
votes
1
answer
746
views
Algebraic structure on homotopy groups of spheres
It is about a "conjecture" I heard (when I was student). There would exist an algebraic structure on the homotopy groups of spheres such that this algebraic structure would be the free algebraic struc …
- 3,901
2
votes
0
answers
95
views
Projective model categories on homotopy equivalent index categories
Consider a fixed proper simplicial combinatorial model category $\mathcal{M}$. Consider a functor $F:I\to J$ between small categories. It induces a right Quillen functor $F^*:\mathcal{M}^J \to \mathca …
- 3,901
5
votes
1
answer
209
views
Model category of diagrams with the colimit detecting the weak equivalences
Let $I$ be a small category and $\mathcal{K}$ be a combinatorial model category.
Is it known a model category structure on the functor category
$\mathcal{K}^I$ such that a map of diagrams $D\to …
- 3,901
6
votes
1
answer
292
views
About the dual of the cube lemma in homotopy theory
Consider the Reedy category $2\rightarrow 1 \leftarrow 0$. Consider a map of diagrams of topological spaces $D\to E$ over this Reedy category:
The maps which are fibrations are depicted with the symb …
- 3,901