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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 ...
K. Strong's user avatar
  • 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 ...
kelly maggs's user avatar
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? ...
Zach Goldthorpe's user avatar
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?
Daniel Bruegmann's user avatar
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 ...
Jonas Linssen's user avatar
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 ...
Andrea Marino's user avatar
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 ...
Giulio Lo Monaco's user avatar
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 ...
Andrea Marino's user avatar
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 ...
Giulio Lo Monaco's user avatar
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 ...
Andrea Marino's user avatar
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 ...
curious math guy's user avatar
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 ...
Giulio Lo Monaco's user avatar
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 ...
F.Abellan's user avatar
  • 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 ...
Emily's user avatar
  • 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 ...
Daniel Gerigk's user avatar
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 ...
Edoardo Lanari's user avatar
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 ...
Edoardo Lanari's user avatar
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 ...
deltmu's user avatar
  • 121