All Questions
Tagged with enriched-category-theory model-categories
12
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
1
answer
408
views
Why do we need enriched model categories?
As far as I understand, model categories mainly provide tools for studying the "homotopy theories" (that is, $\infty$-categories) that are ubiquitous in mathematics. From this point of view, ...
- 6,684
12
votes
1
answer
371
views
What is the right notion of a functor from an internal topological category to a topologically enriched category?
Let $\mathcal{C}$ be a category internal to (some convenient model for) topological spaces (which I will denote by $\mathsf{Top}$). In the question Greg Arone asks:
What is the correct notion of a ...
- 781
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
5
votes
3
answers
534
views
The homotopy category of the category of enriched categories
We know that if $\mathcal C$ is a combinatorial monoidal model category such that all objects are cofibrant and the class of weak equivalences is stable under filtered colimits, then $\mathsf{Cat}_{\...
- 363
3
votes
0
answers
73
views
On cofibrations of simplicially enriched categories
Let $\mathbb{C}$ be an strict 2-category and denote by $C$ is underlying 1-category viewed as as a 2-category only having identity 2-cells.
We have a canonical inclusion functor ,
$$i: C \...
- 457
1
vote
1
answer
171
views
Colocal Objects in Enriched Bousfield Colocalizations
Let $C$ be a $V$-model category, and $\mathcal{K}$ a set of objects of $C$.
Let me denote (derived) simplicial homotopy function complexes by $\text{Dmap}$
and derived $V$-function complexes by $\text{...
14
votes
1
answer
479
views
Why is every object cofibrant in an excellent model category?
In Appendix A.3 of the book higher topos theory appears the notion of an excellent model category (see Definition A.3.2.16). The main feature of this notion is that when $\mathbf{S}$ is an excellent ...
- 9,306
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
views
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 ...
user2664
17
votes
2
answers
1k
views
Model category structure on categories enriched over quasi-coherent sheaves
Gonçalo Tabuada has shown that there is a Quillen model category structure on the category of small dg-categories, i.e. the category of small categories enriched over chain complexes (for a fixed ...
- 538