All Questions
Tagged with enriched-category-theory homotopy-theory
10
questions
2
votes
0
answers
129
views
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 $...
- 3,831
3
votes
1
answer
185
views
Enriched cofibrant replacement in spectrally enriched categories
If $\mathcal{V}$ is a monoidal model category with all objects cofibrant, Theorem 13.5.2. of Categorical Homotopy Theory will guarantee that the functorial cofibrant replacement of a $\mathcal{V}$-...
- 5,511
2
votes
0
answers
142
views
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 ...
- 3,831
4
votes
1
answer
747
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How to compute Homotopy Pullback
What on Earth is a homotopy pullback of
$$A \rightarrow B \leftarrow C \ \ \ \ \ ???$$
Here $A,B,C$ are elements of a category ${\mathcal V}$ enriched in topological spaces (any convenient category ...
- 12.2k
4
votes
0
answers
112
views
Pushforward of an internal category along a functor
Let $F:C\to D$ be a “nice” functor (for example, $H_*(-;\mathbb{Z}):\mathbf{Top}\to \mathbf{Ab}^{\mathbb{Z}}$). Now assume that we have a category $O$ internal to $C$. Is there a canonical way to ...
- 1,623
2
votes
1
answer
824
views
Colimits in the category of simplicial categories
A simplicial category is a category enriched over the monoidal category of simplicial sets (morphism sets are now simplicial sets), and the collection of all such categories forms a category itself (...
- 825
21
votes
2
answers
2k
views
How to stop worrying about enriched categories?
Recently I realized that ordinary category theory is not a suitable language for a big portion of the math I'm having a hard time with these days. One thing in common to all my examples is that they ...
6
votes
1
answer
991
views
Properties of loop space functor from homotopy types to group objects in homotopy types
I am trying to understand some properties of categories enriched in homotopy types, and the following question has become important:
When we take the loop-space of a (connected) homotopy type, we get ...
- 728
2
votes
1
answer
366
views
When are homotopy categories of model categories closed modules over the homotopy category of $(\infty, 1)$-categories?
Let $\mathrm{Quillen}$ be the model category of simplicial sets with the Quillen model structure, and $\mathrm{Joyal}$ the model category of simplicial sets with the Joyal model structure.
As is well-...
- 2,399
0
votes
1
answer
285
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equivalence in simplicial category
Let $(\mathcal{C},W)$ be a category with weak equivalences. One can build from $(\mathcal{C},W)$ its hammock localization $L^{H}(\mathcal{C},W)$ which is a simplicial category $\textit{ie}$ a category ...
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