Questions tagged [simplicial-categories]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
1answer
127 views

Are hammock localizations locally truncated?

Let us take a relative category $(\mathcal{C},\mathcal{W})$, and consider its hammock localization $L_H \mathcal{C}$. It seems to me that for every two objects $X,Y \in \mathcal{C}$ the mapping ...
4
votes
0answers
58 views

Projective objects in the category of simplicial objects in an abelian category

Let $S\mathcal{A}$ be the category of simplicial objects in an abelian category $\mathcal{A}$. In exercise 8.4.5 in Weibel's An Introduction to Homological algebra, it is said that $P \in S\mathcal{A}$...
4
votes
1answer
171 views

when a right lifting property closed under pushouts?

A class of morphisms defined by a right Quillen lifting property (weak orthogonality) is always closed under pullbacks (limits); under what assumptions will it be closed under pushouts (colimits) ? ...
3
votes
0answers
239 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 ...
4
votes
1answer
76 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 ...
4
votes
1answer
120 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
votes
0answers
59 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....
10
votes
1answer
456 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 ...
5
votes
1answer
390 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
votes
1answer
252 views

question about notation in HTT of J.Lurie

In page 27 in HTT of J.Lurie, the expression $$\text{Map}_S(X,Y):=Y^X\times_{S^X}\{\phi\}\in \text{Set}_\Delta$$ appears for simplicial set $X,Y,S$ in Warning 1.2.2.2. However, I couldn't understand ...
2
votes
0answers
112 views

when is the pullback along the shift (decalage) morphism a direct product, in sSets

What is the meaning of the following condition on a morphism in sSets or simplicial topological spaces: a morphism becomes a direct product after the pullback along the shift(decalage) morphism ? ...
2
votes
0answers
37 views

Eilenberg–Zilber-type theorem for Map([n],A), where the degeneracy maps for [n] are forgotten

The following statement should be immediately implied by Eilenberg–Zilber theorem if the sequences $(i_0,\ldots,i_k)$ below are only monotone. But I need the strict monotone version which I believe to ...
6
votes
1answer
444 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 ...
7
votes
1answer
144 views

Locally minimal simplicial categories

Given a (fibrant) simplicially enriched category $\mathcal{C}$, I'm interested in the possibility of replacing it with a weakly equivalent one (in Bergner model structure) such that all the mapping ...
6
votes
0answers
105 views

Morphisms of hammocks in the simplicial localization

Let $\mathcal{C}$ be a category together with a wide subcategory $\mathcal{W} \subset \mathcal{C}$. In Calculating Simplicial Localizations by Dwyer and Kan, a morphism of hammocks is defined ...
1
vote
1answer
124 views

Symetrical simplex category

The simplex category has for objects totally ordered sets $[n]$ , and for morphisms order-preserving functions between those sets. We can see the totally ordered set $[n]$ of size $n$ of the simplex ...
5
votes
1answer
138 views

Cofibrant simplicial categories

Reading about simplicial categories, and in particular the model structure in sCat, I found various sources, among which I can mention this one or section 16.2 in Riehl's book “Categorical homotopy ...
7
votes
1answer
242 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 ...
4
votes
0answers
145 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 ...
3
votes
1answer
183 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 ...
4
votes
0answers
247 views

The k-ification of the compact-open topology for weak Hausdorff compactly generated spaces

Let CGWH be the category of weak Hausdorff compactly generated spaces; see e.g. N.P. Strickland. THE CATEGORY OF CGWH SPACES: Preprint available from https://neil-strickland.staff.shef.ac.uk/courses/...
3
votes
0answers
69 views

A variation of the hammock localization

Let $W \subseteq \mathcal{C}$ be a wide subcategory of a category $\mathcal{C}$. The saturation $\overline{W}$ of $W$ is the class of maps of $\mathcal{C}$ which become isomorphisms in $\mathcal{C}[W^{...
1
vote
0answers
143 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 ...
2
votes
0answers
98 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 ...
2
votes
0answers
51 views

Simplicial models for mapping spaces of filtered maps

Let $S$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. Hence $SIMP(S,K)$ is Kan. Suppose that ${...
3
votes
1answer
103 views

Simplicial models for fibrations between mapping spaces

Let $S,T$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. We let $|S|$ denote the geometric ...
4
votes
3answers
392 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 ...
5
votes
0answers
101 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 \...
6
votes
1answer
234 views

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

I can't understand the proofs of propositions 2.6 and 4.2 in https://www3.nd.edu/~wgd/Dvi/SimplicialLocalizations.pdf We have a category $C$ and a family of maps $W$, and we define the ...
5
votes
1answer
253 views

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

I got very lost in checking that simplicial sets with Quillen model structure are indeed a simplicial model category. Recall that a model category $\mathcal{M}$ is simplicial if it is enriched in $\...
10
votes
1answer
258 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 ...
4
votes
2answers
605 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 ...
3
votes
1answer
381 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 ...
4
votes
1answer
160 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 ...
6
votes
0answers
67 views

Is composition in a simplicially enriched category always determined by a compatible simplicial tensoring (if such exists)?

Let $C$ be a simplicially enriched category, i.e., there are a collection of objects $ob C$, a simplicial set $map_C(X,Y)$ for $X,Y \in ob C$, composition maps $map_C(Y,Z) \times map_C(X,Y) \to map (...
8
votes
0answers
168 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 ...
2
votes
0answers
104 views

Does twisted arrows commute with the simplicial nerve construction?

Let $\mathcal{C}$ be a simplicial category, and let $N(\mathcal{C})$ be its simplicial nerve. We can form the category of twisted arrows as a simplicial category $TwArr(\mathcal{C})$ Now Lurie's ...
3
votes
1answer
264 views

Does $\mathbf{Top}$ admit a simplicial structure

A simplicial category $\mathcal{C}$ is an $\mathbf{sSet}$-enriched category which is bitensored, in that the functors $\mathbf{Map}(-,X)$ and $\mathbf{Map}(X,-)$ admit left adjoints for every $X\in \...
4
votes
0answers
64 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 ...
3
votes
2answers
236 views

Is the projective model structure simplicial?

Let $D$ be a combinatorial simplicial model category (e.g $SSet$ with the standard model structure) and let $C$ be a small simplicial category. Of course, we can consider the projective model ...
2
votes
1answer
253 views

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

May someone provide me the list of basic techniques about simplicial sets, in order to have a good understanding of the definition of a quasicategories, $\infty$-topos, $\infty$-stacks, $\infty$-...
12
votes
2answers
626 views

Simple question: different definitions of Bousfield localization

I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them. First definition: Let $\mathbf{...
2
votes
1answer
188 views

Join of simplicial categories

Let $\mathcal{C},\mathcal{D}$ be simplicial categories. Of course, we have the "naïve" join $\mathcal{C} \star \mathcal{D}$, which has $$ \mathrm{Ob}(\mathcal{C} \star \mathcal{D}) := \mathrm{Ob}(\...
3
votes
1answer
188 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 ...
8
votes
0answers
240 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 ...
4
votes
0answers
519 views

How to compute (co)limits of enriched categories?

I've asked this question on math.stackexchange some time ago (https://math.stackexchange.com/questions/1380176/how-to-compute-colimits-of-enriched-categories) and I received no complete answers, so I'...
5
votes
1answer
482 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{...
1
vote
0answers
178 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....
3
votes
1answer
117 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
vote
1answer
133 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 ...