Questions tagged [simplicial-stuff]

For questions about simplicial sets, simplicial (co)algebras and simplicial objects in other categories; geometric realization, Dold-Kan correspondence, simplicial resolutions etc.

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Equivalence relations: Cosimplicial semilattice?

For $n\ge 0$, let $E_n$ be the set of all equivalence relations on $[n]:=\{0,\dotsc,n\}$. Now given two equivalence relations $R,R'\in E_n$, we build their join $$R\vee R' := \langle R\cup R'\rangle,$$...
14
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1answer
461 views

Simplicial set of permutations

Let $S_k$ be the set of all permutations of $k+1$ elements $0,1,...,k$. introduce boundary maps $d_i : S_k \rightarrow S_{k-1}$ by deleting from permutation $\eta$ element $\eta(i)$ and monotone ...
2
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1answer
187 views

Simplicial manifold associated to Lie groupoid

Let $\Gamma=(\Gamma_1\rightrightarrows \Gamma_0), \Gamma’=(\Gamma’_1\rightrightarrows \Gamma’_0)$ be Lie groupoids and $\Gamma_{\bullet} ,\Gamma’_{\bullet}$ be the simplicial manifolds associated to $\...
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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^{...
4
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102 views

Mysterious identity in cosimplicial $R$-module with Lie brackets

I have a cosimplicial $R$-module $\mathscr{A}:=(A_n)_{n\ge 0}$ and on each $A_n$ a Lie bracket $[-,-]_n:A_n\otimes A_n\to A_n$. Denote the cofaces by $d_i:A_n\to A_{n+1}$ and the codegeneracies by $...
9
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1answer
170 views

On minimal Kan simplicial sets having finite number of simplexes in each dimension

What are the examples of “tame” minimal Kan simplicial sets having finite number of simplexes in each dimension besides simplicial point and $B(G)\approx K(G,1) $ for a finite group $G$? I believe ...
6
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2answers
361 views

Dold-Kan correspondence in the category of symmetric spectra

The Dold-Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes of abelian groups is a classical result. It states that the ...
3
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0answers
48 views

Computing weak operadic colimits as colimits

I am trying to reduce the computation of weak operadic colimits to colimits. Let me introduct some notation. Let $q:C^{\otimes} \to N(Fin_*)$ be a symmetric monoidal category. Let $p: K \to C^{\...
3
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1answer
108 views

Two directed colimits of same spaces with different inclusions

For any natural number $n$, let $i_{n},j_{n}:X_{n}\rightarrow X_{n+1}$ be a pair of monomorphisms of simplcial sets. Define $$X=\operatorname*{colim}_n \{\cdots X_n \rightarrow_{i_n} X_{n+1}\cdots \}...
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0answers
87 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 ...
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79 views

Techniques for computing homotopy pullbacks

I am in the context of simplicial sets. I have a square diagram that I want to show to be homotopy pullback. What I can grasp is that it is a pullback up to some equivalences in the middle of my ...
4
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1answer
111 views

Inner fibrations are Kan fibrations on Map sets

Firstly, a bit of notation. Let $C$ be a simplicial set. We define, for $x,y \in C$ vertices in $C$ $$Map(x,y) = \{x\}\times_{\Delta^{\{0\}}}Map_{sSet}(\Delta^1,C) \times_{\Delta^{\{1\}}} \{y\} $$ ...
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40 views

Some properness condition in simplicial sets

Suppose that $C \to D$ is a trivial Kan fibration, and $D' \to D$ is an equivalence of simplicial sets. Let $C'$ be their pullback. Is it true that $C' \to D'$ is an equivalence? Recall that a ...
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2answers
274 views

What is an example of a quasicategory with an outer 4-horn which has no filler?

A quasicategory has fillers for all inner horns $\Lambda^i[n]$ where $n\geq 2$ and $0<i<n$, but it need not have fillers for $i=0$ (or $i=n)$. In particular, for $n=2$ and $n=3$ there are easy ...
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85 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 ...
3
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1answer
62 views

Contractibility of cocartesian liftings

I am searching to show a quite technical result and I am wondering the following. Suppose $p: C \to D$ is a functor of infinity categories. Take a cell $\Delta^2 \to C$, and suppose that $\Delta^{\{0,...
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196 views

A compendium of weak factorization systems on $sSet$

A (weak) factorization system on a category $\mathcal{C}$ consists of a pair of classes of morphisms in the category $(L,R)$ satisfying Every morphism $f:x \to y \in \mathcal{C}$ can be factored (...
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2answers
431 views

Extending Kan fibrations, without using minimal fibrations

$\require{AMScd}$One thing that needs to be checked to give an interpretation of type theory in simplicial sets (as in Kapulkin-Lumsdaine) is that "the base of the universal fibration is fibrant". ...
2
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1answer
205 views

Simplicial set represented by an (unordered) set

Let $X$ be a (finite if you want) set and form the simplicial set $F^{\bullet}(X)$ with $$ F^{n}(X) = \mathrm{Hom}_{\mathrm{set}} ([n], X) $$ where the right hand side denotes arbitrary maps of sets (...
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51 views

How exactly to adapt Brown's collapse from monoids to algebras?

In The Geometry of Rewriting Systems (Springerlink behind paywall), Kenneth Brown describes a method to collapse the bar resolution of a monoid. Roughly: Given a simplicial set $X$ equipped with a ...
5
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1answer
262 views

pair of injective morphisms of simplicial groups

Let $X$ and $Y$ be two simplicial groups such that there exists $f:X\rightarrow Y$ and $g:Y\rightarrow X $ two injective morphisms of simplicial groups. Suppose that $X$ is contractible, does it ...
5
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1answer
142 views

Simplicial Objects in Additive Categories

I am looking for a reference, preferably as elementary as possible, for the following statement. Let $X_{m,n}$ be a bi-simplicial object in an additive category $\mathcal{A}$. Then the complex $|X_{...
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0answers
49 views

What is the normalized complex of a simplicial set with a monoid action?

This question is a follow up to this question I posted on Math.SE. I will make this question self-contained, though. In a certain point on the paper The Geometry of Rewriting Systems, Kenneth Brown ...
5
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1answer
390 views

A few questions while reading Higher Topos Theory

I am now reading Higher Topos Theory. I have recently met the following questions that I am not able to figure out and I am here to look for some answer or help. First, in Lemma 2.2.3.6, while ...
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1answer
184 views

Thomason fibrant replacement and nerve of a localization

The Thomason model structure on the category $\mathrm{Cat}$ of small categories is transferred along the right adjoint of the adjunction $$\tau_1 \circ \mathrm{Sd}^2 \colon s\mathrm{Set} \...
12
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1answer
350 views

Turning simplicial complexes into simplicial sets without ordering the vertices

Given an abstract simplicial complex $K$, one can make a simplicial set $X(K)$ with $n$-simplices given by sequences $(x_0, \ldots, x_n)$ such that $\{x_0, x_1, \ldots, x_n\}$ is a simplex of $K$. The ...
5
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1answer
151 views

Topological realisation of a stack (explicit description)

Let $X$ be an Artin stack over $k=\mathbf{C}$. I've heard that $X$ has a topological realisation, but I've not been able to find an explicit decription. My first guess would be: take a smooth cover $...
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0answers
66 views

A homotopy problem for morphisms of dg-algebras

Let $\mathfrak g$ be a real finite-dimensional Lie algebra, and suppose we are given two morphisms of dg-algebras $f,g: (C^\bullet(\mathfrak g),d_{CE}) \to (\Omega^\bullet(\Delta^n),d)$, such that ...
6
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142 views

Does Ex^∞ send homotopy inverse limits of ∞-categories to homotopy inverse limits of spaces

Kan's functor $\operatorname{Ex}^{\infty}$ plays the role of total localization or groupoid completion in the theory of $\infty$-categories; specifically, it can be viewed as the left-adjoint of the ...
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2answers
520 views

Why is Kan's $Ex^\infty$ functor useful?

I've always heard that Kan's $Ex^\infty$ functor has important theoretical applications, but the only one I know is to show that the Kan-Quillen model structure is right proper. What else is it useful ...
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46 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
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1answer
66 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 ...
5
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1answer
159 views

Monad, algebras and reflexive coequalizer

Suppose we have an adjunction of categories $F:M\leftrightarrows N:U$. We define the associated (co)monad $G=F\circ U$. For any object $x\in N$ we define the simplicial resolution of $x$ given by $$ ...
6
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1answer
183 views

Explicit chain homotopy for the Alexander-Whitney, Eilenberg-Zilber pair

Let $A$ and $B$ be simplicial abelian groups, and let $N_\ast(-)$ denote the normalized chain complex functor. Let $$AW_{A,B}\colon N_\ast(A\otimes B) \longrightarrow N_\ast(A)\otimes N_\ast(B)$$ and ...
2
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43 views

Contiguity for simplicial maps between simplicial sets

I begin by recalling the definition of contiguous simplicial maps between abstract simplicial complexes: Definition. Two simplicial maps $\varphi,\psi\colon K \to L$ are said to be contiguous if for ...
6
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1answer
133 views

Simplicial localization of the cofibrant-fibrant objects

Let $M$ be a model category. I don't assume that $M$ has functorial factorizations or that $M$ is simplicial. Write $M^{c}$ (respectively, $M^{cf}$) for the full subcategory of $M$ on the cofibrant ...
7
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1answer
244 views

Does the cubical nerve preserve weak equivalences of simplicial sets?

The finite cartesian powers of $\Delta[1]$ form a cubical object in simplicial sets, inducing a "cubical nerve" functor $N_\Box: sSet \to Set^{\Box^{op}}$. $N_\Box$ is a right Quillen equivalence, ...
3
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0answers
183 views

Is there a natural homotopy inverse to the map $∣Sing(X)∣\rightarrow X$

Let $Sing:C \rightarrow SSet$ be the functor sending a CW complex to its singular complex (a simplicial set). Let $∣-∣: SSet \rightarrow C$ be the geometric realization functor. For every $X$ we ...
8
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1answer
196 views

Simplicially enriched cartesian closed categories

In this question I asked whether for a complete and cocomplete cartesian closed category $V$, there can be a complete and cocomplete $V$-category $C$ (with powers and copowers) whose underlying ...
4
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1answer
103 views

Subdivision of simplicial sets but not the barycentric one!

Suppose $K$ and $L$ are simplicial sets. When should one consider that $K$ is a subdivision of $L$? I ask with a view to defining some notion of `finer' generalising that of 'finer triangulation' of ...
3
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0answers
133 views

Why is the Straightening functor the analogue of the Grothendieck construction?

In classical category theory, there is the notion of functor (co)fibered in groupoids. Furthermore, via Grothendieck construction we have an equivalence between peseudo functors into the category of ...
8
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274 views

About Kan-Thurston theorem

The Kan-Thurston theorem obsesses me recently, which is proved by Daniel Kan and William Thurston, Every connected space has the homology of a K(π,1), Topology Vol. 15. pp. 253–258, 1976. Later, there ...
4
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1answer
122 views

Representing simplicial homotopy classes cubically?

Let $(X,x_0)$ be a pointed simplicial set. Assume if you like that $X$ is the nerve of a category but do not assume that $X$ is a Kan complex. Because $Ex^\infty X$ is a Kan complex, every homotopy ...
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0answers
101 views

Maps between simplicial manifolds

Is it true that every continuous simplicial map between smooth simplicial manifolds can be approximate (in some adequate sense) by a smooth simplicial map?
4
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2answers
198 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 ...
3
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0answers
35 views

Could we have the simplicial definition of equivariant derived category of sheaves with arrow direction inversed?

Let $X$ be a topological space and $G$ be a topological group acting on $X$ from the left. We consider the simplicial space $[G\backslash X]_{\cdot}$ where $$ [G\backslash X]_n=\underbrace{G\times \...
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1answer
76 views

Is the map between mapping spaces, induced by the functor $\vert Sing(-)\vert$ continuous?

Let $X$ and $Y$ be topological spaces. Let $\vert Sing(-)\vert$ be the functor which sends a topological space to the (or "a"? there seem to be more possibilites, for me it's just important, that I ...
6
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1answer
405 views

Need help understanding comment in Higher Topos Theory

I am having trouble understanding this part in Lurie's Higher Topos Theory. This can be found in section 2.4.4.4 right after Lemma 2.4.4.1. Lemma 2.4.4.1. Let $p : \mathcal{C} \rightarrow \mathcal{...
11
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1answer
248 views

Which maps of simplicial sets geometrically realize to fibrations?

If $f:X\to Y$ is a Kan fibration of simplicial sets, then its geometric realization $|f| : |X|\to |Y|$ is (in some suitable convenient category of topological spaces, like compactly generated ones) a ...
2
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1answer
78 views

Factorization of a map from a contractible Kan complex through a Kan complex

Suppose we are given a contractible Kan complex $S$ and a map of simplicial set $f : S \rightarrow T$. Under what conditions can we say that $f$ factors through the largest Kan complex $Z$ contained ...