Questions tagged [simplicial-categories]

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Defining null-homotopy in terms of endofunctors in an arbitrary simplicial category

I am looking for references defining contractible or null-homotopic in terms of endofunctors $\Delta\to\Delta$. Let $[+1]:\Delta\to\Delta, n\mapsto n+1$ be the endofunctor of $\Delta$ adding a new ...
4 votes
1 answer
129 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 ...
0 votes
1 answer
268 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 ...
0 votes
1 answer
154 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 ...
2 votes
0 answers
102 views

Are homotopy colimits strict?

Let's say we are working with a fibrant simplicially enriched category $\mathbf{B}$ that has all limits and all homotopy limits, and let $\mathbf{A}$ be a full subcategory that is closed under weak ...
3 votes
1 answer
200 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 ...
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1 vote
0 answers
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combinatorial equivalence of abstract simplicial complexes

what is the most natural equivalence one defines on abstract simplicial complexes ? The definition is purely combinatorial of abstract SCs. It seems to me that the combinatorial equivalence is however ...
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1 vote
0 answers
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Cohomology of bisimplicial set is the cohomology of the total simplicial set?

Given a bisimplicial manifold (or set, topological space) there is the bar construction, which assigns to it a simplicial manifold, called the total simplicial manifold. Now stepping back for a moment,...
1 vote
0 answers
55 views

Tensored and cotensored simplicial comma category

To transfer a tensored and cotensored simplicially enriched structure from a category $\mathcal{C}$ to $(\mathcal{C}\downarrow Z)$, we define $(X\to Z)\otimes K$ by the composite $(X\otimes K \to X \...
3 votes
2 answers
127 views

The exactness of the associated chain complex of a simplicial free abelian group over a finite set and the normalization theorem

Update: Now I know why my method fails. But I still wanna know how to work out the original question, that is to show the exactness of the chain complex $C_*(X)$ except for two positions $n=0,N-1$. ...
3 votes
0 answers
78 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?
8 votes
1 answer
238 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 ...
2 votes
1 answer
158 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
0 answers
77 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 $P \in S\mathcal{A}$...
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5 votes
1 answer
361 views

When is 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)? In ...
6 votes
0 answers
419 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 ...
5 votes
1 answer
91 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
1 answer
137 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
0 answers
60 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....
12 votes
1 answer
604 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
1 answer
477 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
1 answer
262 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 ...
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2 votes
0 answers
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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
0 answers
42 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
1 answer
480 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
1 answer
149 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
0 answers
111 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 ...
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1 vote
1 answer
127 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 ...
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6 votes
1 answer
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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
1 answer
281 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
0 answers
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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
1 answer
206 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 ...
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4 votes
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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
0 answers
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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^{...
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1 vote
0 answers
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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
0 answers
119 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
0 answers
52 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
1 answer
125 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
3 answers
453 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 ...
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5 votes
0 answers
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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 \...
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6 votes
1 answer
239 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 ...
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5 votes
1 answer
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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
1 answer
264 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 ...
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4 votes
2 answers
698 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 ...
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3 votes
1 answer
463 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 ...
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4 votes
1 answer
166 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 ...
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6 votes
0 answers
69 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
0 answers
174 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
0 answers
108 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
1 answer
284 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 \...
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