All Questions
Tagged with homotopy-theory simplicial-categories
25 questions
5
votes
1
answer
157
views
Simplicial objects in quasicategory which come from homotopy coherent nerve
Let $\mathcal{C}$ be a simplicially enriched category whose Hom-objects are all Kan complexes. Denote by $N\mathcal{C}$ the homotopy-coherent nerve of $\mathcal{C}$, which is a quasicategory. Suppose ...
- 423
4
votes
1
answer
174
views
Classifying the endofunctors of the category $\Delta$ of finite linear orders
Is there a theory of endofunctors of the category $\Delta$ of finite linear orders ?
Can they be classified ? Is there a reference on this ?
Can one classify endofunctors $T:\Delta\to\Delta$ which ...
- 317
0
votes
1
answer
288
views
Defining homotopy via the “doubling” endofunctor of a simplicial category
I am looking for a reference explicitly defining simplicial homotopy
in terms of endofunctors of $\Delta$, and developing homotopy theory in this terms.
The following is a particular question.
Is it ...
- 317
1
vote
1
answer
186
views
Defining homotopy via endofunctors of a simplicial category
$\newcommand\sSets{\text{sSets}}\newcommand\sing{\text{sing}}\DeclareMathOperator\Hom{Hom}\newcommand\Top{\text{Top}}$I am looking for a reference describing the notions of homotopy and ...
- 317
3
votes
1
answer
372
views
Homotopy coherent nerve versus simplicial nerve
Background
Recently I asked a question on a particular construction ot the classifying space of a topological group. I got an answer, but it relied on nontrivial Quillen equivalences between various ...
- 2,292
7
votes
1
answer
359
views
How do the various homotopy 2-categories compare?
There are various models of $\infty$-categories floating around, so there are as many models of the associated homotopy 1- and 2-categories. Because the relations between the former are worked out in ...
- 225
2
votes
0
answers
65
views
homotopy coherent G-action on tensor product of complexes
Let $G$ be a discrete group and $k$ a field. Suppose $C_1$ and $C_2$ are complexes over $k$ with homotopy coherent actions of $G$ in the sense of Cordier (I've been reading https://arxiv.org/pdf/1801....
- 101
5
votes
1
answer
659
views
Homotopy coherent colimits in chain complexes
In remark 1.2.6.2 (HTT), Lurie states that
Another possible approach to the problem of homotopy
coherence is to restrict our attention to simplicial (or topological) categories
C in which every ...
- 2,152
4
votes
0
answers
164
views
Is simplicial localization part of a Quillen equivalence between relative categories and simplicial categories?
There are many models for $\infty$-categories. One of them is relative categories – AKA categories with weak equivalences – which have a model structure due to Barwick–Kan. Another one is simplicial ...
- 5,040
1
vote
0
answers
214
views
Existence of tensor product of infinity operads
I am trying to show, or find a reference, for the following fact:
"Given O,P two infinity operads [in the sense of Lurie, HA, Definition 2.1.1.10], there always exist a tensor product".
In other ...
- 2,152
2
votes
0
answers
170
views
Homotopy colimits of simplicial objects
Given a simplicial combinatorial model category $\mathcal{M}$ and a simplicial diagram $F\colon \Delta^{\mathrm{op}} \rightarrow \mathcal{M}$, is there a nice (i.e. explicitely computable) way of ...
- 1,403
5
votes
0
answers
124
views
When is a bisimplicial set diagonal fibrant
Let $sSet^2$ be the category of bisimplicial sets.
In the diagonal model structure on $sSet^2$ weak equivalences are diagonal weak equivalence (i.e.$ X \rightarrow Y$ is a weak equivalence if $dX \...
- 263
10
votes
1
answer
288
views
Criterion for homotopy pullback square of simplicial categories
Assume given a pullback square of simplicial categories
$$\begin{array}[c]{ccc}
A&{\rightarrow}&B\\
\downarrow&&\downarrow\\
C&{\rightarrow}&D.
\end{array}$$
Suppose further ...
- 660
4
votes
2
answers
855
views
Filtered colimit of fibrations
In a model category $\mathcal{C}$, is the filtered colimit of fibrations, resp. trivial fibrations, a fibration, resp. trivial fibration?
Thm. 1.2.3.5 in Toen-Vezzosi's "Homotopical algebraic ...
user95222
3
votes
1
answer
660
views
Coproducts of weak equivalences
In a model category $\mathcal{C}$ admitting a forgetful functor to simplicial sets, is the coproduct of weak equivalences a weak equivalence?
Say even just coproducts indexed by $\mathbf{N}$.
A ...
user95222
4
votes
1
answer
193
views
Kan complexes and semigroups
Given a simplicial commutative semigroup:
(1) is it true that its underlying simplicial set is a Kan complex if and only if the simplicial semigroup was a simplicial group?
(2) is the constant ...
user92332
8
votes
0
answers
190
views
Unaugmentable cosimplicial simplicial sheaves and realization functor
I'm studying the construction of the $\mathrm{Sing}$ functor in Morel-Voevodsky ``$\mathbb{A}^1$-homotopy theory of schemes'' and I was trying to understand the properties of its left adjoint, the ...
- 303
4
votes
0
answers
76
views
In which sense is the relativization functor "preferred"?
In A characterization of simplicial localization functors and a discussion of DK equivalences Barwick and Kan state that, while there is no preferred localization functor from relative categories to ...
- 651
6
votes
1
answer
613
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
1
vote
0
answers
231
views
Internal Hom on simplicial presheaves and the preservation of cofibrant objects
1)Let $\mathcal{C}$ be a cartesian closed small category. Let $\operatorname{Map}\: : \: sPsh(\mathcal{C})\times sPsh(\mathcal{C})\to sPsh(\mathcal{C})$ be the internal Hom of simplicial presheaves, i....
- 1,424
3
votes
1
answer
147
views
Segal maps for Segal precategories
A Segal precategory is just a simplicial space $X:\Delta^{op} \to sSet$ such that its $0$-th space is discrete (i.e. constant). A Segal category is defined everywhere in the literature as a Segal ...
- 1,403
1
vote
1
answer
153
views
Homotopy invariance of Kan nerve of simplicial categories
The following question concerns the well-known paper of Dwyer and Kan "Localization of Simplicial Categories". They define a nerve for simplicial categories (with fixed set of objects $O$), by the ...
- 1,403
0
votes
1
answer
293
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
1
vote
1
answer
348
views
Homotopy limit of a cosimplicial category
Consider the usual model structure on Cat (category of small categories).
Which are the fibrations of the injective model structure on the category of cosimplicial categories $Fun( \Delta ,Cat )$?
...
- 875
5
votes
0
answers
335
views
Simplicial localisation and infinity categories
If $(\mathcal{C},W)$ is a category with weak equivalences then we may naturally form its Dwyer-Kan simplicial localisation $L(\mathcal{C}, W)$. This is a simplicial category which naturally gives a ...
- 121