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 ...
- 30.3k
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 ...
- 15.8k
12
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 ...
- 13.6k
11
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 ...
- 27.5k
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
$$ \text{Map}_S(X,Y) = \{f\colon X\to Y: \psi f=\...
- 56.9k
8
votes
Accepted
How do the various homotopy 2-categories compare?
The simplicial sets $h_2(N^\Delta(\mathcal{C}))$ and $N^D(H_2(\mathcal{C}))$ are isomorphic. To prove this, observe that the universal property of $h_2(N^\Delta(\mathcal{C}))$ applied to the image ...
- 5,539
7
votes
Accepted
How are simplicial sets with Quillen model structure a simplicial model category?
The trick is to check that the corner map $$\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 \...
- 19.6k
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 ...
- 1,569
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 ...
- 9,263
5
votes
Accepted
Is hammock localization a localization in the sense of Lurie?
It's generally best not to leave questions without an answer, even if they are answered in the comments. MO best practice is to post a CW answer summarizing the answer from the comments. In this case, ...
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....
- 9,481
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 ...
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{\...
- 11.8k
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 ...
- 63.9k
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 ...
- 63.9k
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 $[-...
- 63.9k
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.
...
- 15.8k
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 ...
- 37.8k
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 ...
- 37.8k
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 ...
- 66.8k
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 ...
- 6,406
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 $\Delta^n\to S$. ...
- 16.5k
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 ...
- 9,263
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” ...
- 37.8k
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 $D$ ...
- 37.8k
2
votes
Does $\mathbf{Top}$ admit a simplicial structure
Yes, it does, if by Top you mean compactly generated spaces (as Dylan pointed out). Then, the internal hom is a space (with the compact-open topology), and its kification is a k-space satisfying hom-...
- 30.3k
2
votes
Cofibrant simplicial categories
Simplicial categories $[n]$ are indeed cofibrant. The Simplicial category $\mathfrak{C}\Delta^n$ is also cofibrant and weakly equivalent to $[n]$, so the former is a cofibrant replacement of the ...
- 4,459
2
votes
Non-enriched Bousfield localizations
This is not a full answer, but too long for a comment.
Here's a relevant paper. Mazel-Gee proves the folklore claim that a Quillen adjunction induces an adjunction on underlying $\infty$-categories ...
- 15.8k
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 $n \geq 3$.
...
- 27.5k
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