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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
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 ...
user420620's user avatar
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 ...
user420620's user avatar
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 ...
user420620's user avatar
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 ...
Ken's user avatar
  • 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 ...
Jonas Linssen's user avatar
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....
mathdonkey's user avatar
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 ...
Andrea Marino's user avatar
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 ...
Najib Idrissi's user avatar
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 ...
Andrea Marino's user avatar
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 ...
Edoardo Lanari's user avatar
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 \...
Girish's user avatar
  • 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 ...
Edouard's user avatar
  • 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 ...
user avatar
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 ...
user avatar
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 ...
user avatar
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 ...
Stefano Nicotra's user avatar
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 ...
Adrian Clough's user avatar
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{...
Marc Nieper-Wißkirchen's user avatar
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....
Cepu's user avatar
  • 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 ...
Edoardo Lanari's user avatar
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 ...
Edoardo Lanari's user avatar
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 ...
user avatar
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 $Fun( \Delta ,Cat )$? ...
Fernando's user avatar
  • 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 ...
deltmu's user avatar
  • 121