# Questions tagged [simplicial-categories]

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69
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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 ...

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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 ...

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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 ...

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$\newcommand\sSets{\text{sSets}}\newcommand\sing{\text{sing}}\DeclareMathOperator\Hom{Hom}\newcommand\Top{\text{Top}}$I am looking for a reference describing the notions of homotopy and ...

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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 ...

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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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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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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,...

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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 \...

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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 $n=0,N-1$.
...

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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?

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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 ...

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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 ...

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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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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 ...

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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 ...

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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 ...

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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 ...

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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....

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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 ...

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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 ...

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262
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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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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 ?
...

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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 ...

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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 ...

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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 ...

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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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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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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 ...

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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 ...

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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 ...

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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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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/...

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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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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 ...

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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 ...

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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 ${...

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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 ...

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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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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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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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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 $\...

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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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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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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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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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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 (...

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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 ...

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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 ...

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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 \...