All Questions
Tagged with simplicial-categories infinity-categories
18 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
3
votes
1
answer
133
views
$n$-truncation of a Simplicial Model Category
I'm working in the category of rational $CDGAs$ and trying to find a reference/construction of a natural $2$-categorical structure via truncation of the mapping spaces.
In my head, the key point is ...
- 285
3
votes
0
answers
55
views
Recognising absolute distributors in terms of simplicial model categories
Briefly, my question is the following:
Can we recognise when a simplicial model category $\def\cM{\mathcal M}\cM$ is an absolute distributor, using only the language of (simplicial) model categories?
...
- 627
3
votes
0
answers
99
views
Singular complex and homotopy coherent nerve as simplicial sets
Let $X$ be a CW complex.
Is the simplicial set $\ \mathrm{Sing}\ X$ isomorphic to the homotopy coherent nerve of some Kan enriched category?
Is this true for $X$ = the real line?
- 890
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
8
votes
1
answer
860
views
Is hammock localization a localization in the sense of Lurie?
In a series of papers ([1], [2] and [3]), Dwyer and Kan introduced the hammock localization [2] as an effective technique to compute the simplicial localization of a model category [1]. This is meant ...
- 2,152
6
votes
1
answer
113
views
Non-enriched Bousfield localizations
We know that whenever we have a Bousfield localization between two simplicial model categories, this gives rise to a reflective subcategory in $\infty$-categories (or coreflective, depending on the ...
- 1,071
4
votes
1
answer
191
views
Homotopy coherent space maps induces homotopy coherent chain complex morphisms
It is an elementary fact that a map $f:X \to Y$ between spaces induces a chain complex morphism $f_* : C_*(X) \to C_*(Y) $. This allows one to transfer 1-category theoretic arguments from spaces to ...
- 2,152
12
votes
1
answer
894
views
Modern proofs for simplicial localizations
I know that the references usually regarded as standard for simplicial localizations are the Dwyer and Kan's three articles from the 80's. I would be interested in a more modern approach to the ...
- 1,071
5
votes
1
answer
658
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
6
votes
1
answer
540
views
"Universal" triangulated category
Let $\mathcal{C}$ be some category. One way to map this category into a triangulated category is to take the category of simplicial objects $s\mathcal{C}$ (which is an $\infty$-category), take its ...
- 4,386
8
votes
1
answer
335
views
Is the simplicial nerve a localization?
Given a simplicial category $\mathcal{C}_{\ast}$ (if necessary, you may assume it's fibrant), denote as $\mathcal{C}$ its underlying ordinary category, and as $\mathcal{W}$ the class of all ...
- 1,071
3
votes
1
answer
264
views
Rigidification of marked simplicial sets
It is well known that there exists a Quillen equivalence,
$$\mathfrak{C}: Set_{\Delta} \rightleftarrows Cat_{\Delta}: N, \enspace \mathfrak{C} \dashv N$$
between Joyal's model structure on ...
- 457
4
votes
3
answers
557
views
Proof that a quasicategory is equivalent to the homotopy coherent nerve of a simplicial category
On the abstract of a paper by Emily Riehl and Dominic Verity, it is stated that
Every quasicategory arises as a the homotopy coherent nerve of a simplicial category up to equivalence.
Where can ...
- 11.8k
3
votes
1
answer
217
views
explicit description of the cosimplicial simplicial set $Q^{\bullet}$
I'm struggling to understand the explicit description of the cosimplicial simplicial set $Q^{\bullet}$ on page 76 (section 2.2.2) of Lurie's book Higher Topos Theory, and would be grateful if someone ...
- 811
8
votes
0
answers
263
views
A completeness criterion for $\infty$-categories
We all know that for ordinary categories $\mathscr{C}, \mathscr{D}$ (with $\mathscr{C}$ small) the limit of a functor $F:\mathscr{C} \to \mathscr{D}$, if it exists, can be constructed by using ...
- 1,403
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
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