14
votes

Accepted

### "Universal" triangulated category

I will give a partial answer. I note that the OP has asked a LOT of questions recently (I count 12 so far in the first 9 days of August), and many of them are good questions on which much research has ...

13
votes

Accepted

### Homotopy coherent colimits in chain complexes

The result is not only true for simplicial model categories, but for plain combinatorial model categories too - this is Higher Algebra 1.3.4.25..
In fact, for this you can reduce to the case of ...

10
votes

Accepted

### Is the simplicial nerve a localization?

This is not true. Here is a counter example. We let $\mathcal{C}_*$ be the following simplicial category. It has two objects 0 and 1. Their only endomorphisms are the identity. There are no morphisms ...

10
votes

Accepted

### Modern proofs for simplicial localizations

For Question 1. It is documented in Corolary 4.2.4.8 in Lurie's Higher Topos theory that $N_\Delta(\underline{M}^{cf})$ is an $\infty$-category with small limits (and small colimits). Moreover, the ...

9
votes

Accepted

### question about notation in HTT of J.Lurie

Just think in terms of ordinary sets for the moment. We have sets and maps $X\xrightarrow{\phi}S\xleftarrow{\psi}Y$ and we want to think about the set
$$ \text{Map}_S(X,Y) = \{f\colon X\to Y: \psi f=\...

8
votes

Accepted

### How do the various homotopy 2-categories compare?

The simplicial sets $h_2(N^\Delta(\mathcal{C}))$ and $N^D(H_2(\mathcal{C}))$ are isomorphic. To prove this, observe that the universal property of $h_2(N^\Delta(\mathcal{C}))$ applied to the image ...

7
votes

Accepted

### How are simplicial sets with Quillen model structure a simplicial model category?

The trick is to check that the corner map $$\lambda^n_k\bar{\times}\delta^m:\Lambda^n_k \times \Delta^m \coprod_{\Lambda^n_k\times \partial \Delta^m} \Delta^n \times \partial \Delta^m \hookrightarrow \...

6
votes

Accepted

### Proof that a quasicategory is equivalent to the homotopy coherent nerve of a simplicial category

Sorry, we should have made this more clear. One proof appears as Theorem 7.2.2 in our previous paper in this series, The comprehension construction. As suggested by others, it follows from a suitably ...

6
votes

### Classifying the endofunctors of the category $\Delta$ of finite linear orders

See Edgewise subdivision and simple maps by Knut Berg (supervised by me), Generalized edgewise subdivisions by Katerina Velcheva (supervised by Clark Barwick) and the earlier MathOverflow question ...

5
votes

### Simple question: different definitions of Bousfield localization

There are many equivalent ways of defining the local equivalences. Let $\mathcal{M}$ be a simplicial model category with a cofibrant replacement functor $Q : \mathcal{M} \to \mathcal{M}$. Then, for a ...

5
votes

Accepted

### Criterion for homotopy pullback square of simplicial categories

Yes. In fact such a square can be replaced with a weakly equivalent Reedy fibrant pullback square without changing the object set of any of the simplicial categories. For a proof see, e.g., Lemma 3.1....

4
votes

Accepted

### Homotopy coherent space maps induces homotopy coherent chain complex morphisms

Be careful with this "simplicial structure on chain complexes". It's not really well-defined, as discussed in the comments below my answer here. Also see my remark at the end of this post.
...

4
votes

Accepted

### Simplicial models for fibrations between mapping spaces

Yes, these agree.
The usual model structures on $C = sSet$ and on $C = Top$ are both cartesian monoidal. So the functor $[-,X] : C^{op} \to C$ is a right Quillen functor when $X$ is fibrant (where $[-...

4
votes

Accepted

### Locally minimal simplicial categories

It's not possible in general to ensure that all the hom-spaces in a simplicial category are minimally fibrant. Here's a counterexample inspired by Isbell.
Consider $Set$ with its cartesian monoidal ...

4
votes

### Proof that a quasicategory is equivalent to the homotopy coherent nerve of a simplicial category

The result is actually valid for any simplicial set.
Lemma. If
is a Quillen equivalence, then the composition
$$A\xrightarrow{\eta_A}G(F(A))\xrightarrow{G\left(P_{F(A)}\right)}G(P(F(A)))$$
is a weak ...

Community wiki

4
votes

### Proof that a quasicategory is equivalent to the homotopy coherent nerve of a simplicial category

Let $\mathbf{C}$ be a quasicategory. Using a version of the Yoneda lemma for quasicategories, Joyal constructs, in Section 15.3 of his Notes on Quasi-Categories, a simplicial category $\overline{\...

4
votes

### Understanding two proofs in Dwyer and Kan article "Simplicial Localizations"

It's not so bad to prove 4.2 directly in terms of generators and relations. Dwyer and Kan define $LC = F_\ast C [F_\ast W^{-1}]$ where $F_n$ is the $(n+1)$st iteration of the free category comonad. So ...

4
votes

Accepted

### Simple question: different definitions of Bousfield localization

Yes, they are the same. In order to prove it, we should show that they have the same new weak equivalences (this is enough, because both have the same cofibrations). Have a look at Barwick's paper On ...

4
votes

Accepted

### explicit description of the cosimplicial simplicial set $Q^{\bullet}$

Here is how I think about this; don't know if it will help.
Start with the straightening construction: this takes a map $f\colon X\to S$ of simplicial sets to a simplicial functor $\def\St{\mathrm{St}...

4
votes

### Defining homotopy via the “doubling” endofunctor of a simplicial category

Once we correct the definition of $[×2]$ to make it a functor (currently it does not preserve identities) by setting $f'=f⊔f$, it is true that the construction described in the main post gives rise to ...

3
votes

### Basic technical things about simplicial sets to have a good understanding of quasicategories

Simplicial sets are absolutely fundamental to the study of quasi-categories. There's no easy list of things you have to know and other things you don't have to know. Therefore, you should try to gain ...

3
votes

### Is the projective model structure simplicial?

The general case is noted in A.3.3.4 in Higher topos theory. Actually it's quite easy to prove, using the characterization of simplicial model categories in terms of powers (cotensors), since the ...

3
votes

Accepted

### Kan complexes and semigroups

(2) is true (and so (1) is false).
To see it, note that every horn $\Lambda^n_i\to S$ to a constant simplicial set must be constant, and so it can be filled by the constant horn $\Delta^n\to S$. ...

3
votes

Accepted

### Symetrical simplex category

You haven't specified what the morphisms in your "symmetric simplex category" are supposed to be, and there are two natural choices:
Functors, or
Natural isomorphism classes of functors.
In the ...

3
votes

Accepted

### Homotopy coherent nerve versus simplicial nerve

Both $\def\W{{\bar W}}\W$ and $\def\N{\mathfrak{N}}\N$ are right Quillen functors from the model category of simplicial groups to the model category of reduced simplicial sets (see the original paper ...

3
votes

### Defining homotopy via endofunctors of a simplicial category

There is a problem with base points in your claim. If $X$ is discrete, then ${\rm sing} X_\bullet$ and ${\rm sing} X_\bullet \circ [+1]$ are the same, so every map from ${\rm sing} F_\bullet$ factors ...

3
votes

Accepted

### Hammock localization and free adjoints

A connection between Dwyer–Kan hammock localizations and adjoints in 2-categories certainly exists. As already mentioned in the comments, as early as 2002, Dawson–Paré–Pronk in “Adjoining adjoints” ...

3
votes

Accepted

### Simplicial objects in quasicategory which come from homotopy coherent nerve

It is easy to construct counterexamples already for the full subcategory $\{[0],[1]\}⊂Δ$. The category $\cal C$ can be constructed by applying the nerve functor to hom-objects of a category $D$ ...

2
votes

Accepted

### Are hammock localizations locally truncated?

Your calculation is correct. For every two objects $X, Y \in \mathcal{C}$, the hom space $L^H\mathcal{C}(X,Y)$ has the right lifting property against $\partial \Delta^n \to \Delta^n$ for $n \geq 3$.
...

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