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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.
2
votes
0
answers
133
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Tensor product of objectwise weak homotopy equivalences of $\mathcal{M}$-spaces
I consider the enriched category $[\mathcal{M}^{op},\mathrm{Top}]$ of enriched functors (I call them $\mathcal{M}$-spaces) from the enriched small category $\mathcal{M}^{op}$ to the enriched category …
- 3,901
2
votes
0
answers
92
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Explicit CW-complex replacement of the space of reparametrization maps
Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-co …
- 3,901
2
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0
answers
95
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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
3
votes
0
answers
98
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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
5
votes
1
answer
173
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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 …
- 66.8k
2
votes
0
answers
142
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When this coend is invariant up to homotopy?
It is a follow-up of my question Calculation of the homotopy colimit of a diagram of spaces which was badly formulated.
Consider a fixed diagram $D:I^{op}\to {\rm Top}$ where ${\rm Top}$ is
a conveni …
- 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
6
votes
3
answers
525
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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 …
- 30.3k
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 …
- 15.9k
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
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
1
vote
On model categories where every object is bifibrant
Look at this paper for some examples: Strong cofibrations and fibrations in enriched categories by R. Schwänzl and R. M. Vogt.
- 3,901
6
votes
0
answers
232
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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
2
votes
0
answers
130
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Enriched homotopy colimit and space of paths
I work in the category of $\Delta$-generated spaces. Let $G$ be the group of strictly increasing continuous bijections from $[0,1]$ to itself (they necessarily take $0$ to $0$ and $1$ to $1$). I call …
- 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 …
- 1,124