The simplicial-stuff tag has no usage guidance.

**6**

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+200

### Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar setting)

Given a fibration $p : Y \to X$ in simplicial sets (or any other model category), there are various ways to construct its fibrewise suspension, i.e. its suspension as an object of the slice ...

**4**

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**1**answer

148 views

### Basic questions about simplicial commutative rings

I am trying to learn about simplicial commutative rings, and would be grateful if one can help with some basic facts about them. Basically, I would like to understand how to do homological algebra ...

**4**

votes

**1**answer

102 views

### For a simplicial set $X$, is the category of non-degenerate simplices of $X$ a full subcategory of the category of simplices of $X$?

I'm slightly confused, but I think you can help me. Let $X$ be a simplicial set. The category of simplices of $X$ and its subcategory of non-degenerate simplices are defined at ...

**7**

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**0**answers

118 views

### $E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral ...

**3**

votes

**1**answer

210 views

### Does the fat geometric realization take limits to homotopy limits?

I am deeply confused about geometric realizations and finite limits.
Suppose that I am working with simplicial sets (I dont need simplicial spaces) that are "good" in the sense of Segal so that I can ...

**2**

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**0**answers

90 views

### Is the dg-nerve functor a Quillen equivalence?

Lurie defines the dg-nerve $N_{dg}(\mathcal{C})$ of a dg-category $\mathcal{C}$ in Higher Algebra Construction 1.3.1.6: for each $n \geq 0$, we define $N_{dg}(\mathcal{C})_n\simeq ...

**1**

vote

**1**answer

109 views

### Does a topological hypercover always have free degeneracies?

This question arises when I am reading Dugger and Isaksen's "Hypercovers in topology". According to Definition 4.1 in that paper, A hypercover of a space $X$ is an augmented simplicial space $U_*\to ...

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133 views

### Why should we regard $PL(M)$ as a simplicial group?

Let $M$ be a manifold. If $M$ is smooth, it is clear what $\text{Diff}(M)$ should be, namely it should be the set of diffeomorphisms of $M$ equipped with the topology in which a sequence of ...

**4**

votes

**1**answer

188 views

### Is the simplicial set $(\Delta_3/\partial \Delta_3)^{\Delta_1}$ finite?

A simplicial set is called finite if it has only a finite number of non degenerate simplicies (or equivalently, if its number of $n$-simplicies grows polynomially with $n$). In an answer to a ...

**2**

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**1**answer

156 views

### Equivalent definition of a Kan fibration

It is known (follows for example from Proposition 4.2 of Simplicial Homotopy Theory by Goerss and Jardine) that a Kan-fibration can be defined as a map having the right lifting property with respect ...

**4**

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**1**answer

274 views

### Natural transformations induce homotopies - Is this true in the “fat” world?

Let $\mathcal{C}, \mathcal{D}$ be categories internal to topological spaces (or compactly generated Hausdorff spaces, if you like) $F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ be continuous functors ...

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**0**answers

155 views

### Does the fat realization of simplicial spaces commute with finite limits up to homotopy?

I'm willing to work in the category of compactly generated Hausdorff spaces.
The ordinary geometric realization of a simplicial space commutes with taking pullbacks and preserves the terminal object, ...

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**0**answers

60 views

### Does it require Reedy fibrancy when we want the totalization to be weakly equivalent to the homotopy limit?

This question arises when I am reading the last two Chapter of Hirschhorn's "Model categories and their localizations"
In Part (2) of Theorem 19.8.4 of that book it says
If ...

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72 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, ...

**2**

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**1**answer

262 views

### Building $(\infty,2)$-categories from $\infty$-categories

Let $Y$ be a marked simplicial set, whose underlying simplicial set is also denoted by $Y$. Let $X$ be a scaled simplicial set such that the decalage of its underlying simplicial set is $Y$. $X$ is ...

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**3**answers

90 views

### smash product of pointed spaces preserve weak equivalences

Consider the category of pointed simplicial sets with usual notion of weak equivalence. The question is does the functor
$$Y \mapsto X\wedge Y$$ preserve weak equivalences? or at the very least does ...

**3**

votes

**1**answer

90 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 ...

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110 views

### Homotopy pullback preserving functor

In the paper http://arxiv.org/pdf/math/0101162.pdf, the authors claim during the proof of Prop. 4.2 that a functor $F:A \to B$ which preserves fibrations and weak equivalences preserves homotopy ...

**11**

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**1**answer

329 views

### Verifying that $\epsilon^!$ is indeed the right adjoint of $\epsilon_*$ in the context of algebraic stacks

The question is about the last paragraph of Remark 12.4.3 in the book on algebraic stacks by Laumon and Moret-Bailly.
Let $S$ be a (quasi-separated) scheme and let $\mathscr{X}$ be an algebraic stack ...

**4**

votes

**2**answers

298 views

### The classifying space of an infinite totally ordered set is contractible

I asked this question on math.stackexchange, but no one answered.
Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This ...

**5**

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**1**answer

386 views

### Generalize $\pi_0(B\mathcal{C})\cong\{\text{objects}\}/\{\text{morphisms}\}$ to categories internal to topological spaces

Warmup:
Let $\mathcal{C}$ be an ordinary category. Denote by $$B\mathcal{C}=(\coprod_{i\in\mathbb{N_0}}N_{i}(\mathcal{C})\times\Delta^i)/\tilde{}$$ its classifying space, i.e. the geometric ...

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**1**answer

118 views

### $\pi_0$ of a cosimplicial space

Let $n\mapsto X^n$ be a cosimplicial simplicial set and $X:= \underset{\longleftarrow}{\rm holim}\ X^n$ the homotopy limit. Is the natural map
$$ \pi_0(X) \to \underset{\longleftarrow}{\rm lim}\ ...

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**0**answers

48 views

### Contraction of simplicial presheaves

Let $X,Y$ be two simplicial presheaves on a small category $\mathcal{C}$, let $*$ be the final simplicial presheaf. Consider the category of simplicial presheaves equipped with its projective model ...

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**1**answer

172 views

### Technical lemma about frame and cosimplicial resolution

I'm reading the Hirschhorn's book model categories and their localization and I have a question about frames and resolutions.
Following the book (definition 16.6.1) a cosimplicial frame on an object ...

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**1**answer

95 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 ...

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votes

**1**answer

80 views

### Monoidal structure on simplicial sheaves

Let $\mathcal{C}$ be a site and let $sPh(\mathcal{C})_{proj}$ be the category of simplicial presheaves equipped with the projective model structure. This category is a closed monoidal model category ...

**5**

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**0**answers

181 views

### Is there a higher, “orientalish” version of geometric realisation?

Geometric realisation of simplicial sets can be roughly thought of like this:
In some category $\mathcal{C}$, we choose an object for every abstract $n$-simplex. In topological spaces, we would ...

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votes

**1**answer

197 views

### Glueing a property via homotopy colimits

I have a problem concerning a fact which is stated without proof in this Rezk's draft: http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf .
In the proof of Lemma 2.11, we are given a ...

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**1**answer

406 views

### Continuous maps to fat geometric realizations of simplicial spaces

The nLab page on partitions of unity mentions the application of partitions of unity as a way to construct continuous maps to geometric realizations of simplicial spaces. However I often feel ...

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**0**answers

26 views

### Criterion for projectively cofibrant diagrams? [duplicate]

(repost from math.stackexchange)
Let $\mathbf{D}$ be a small category, $\mathbf{Set}$ the category of sets and $\mathbf{sSet}$ the category of simplicial sets with its usual model structure. The ...

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**1**answer

88 views

### Does the kan extension preserves contractible presheaves?

Let $\mathcal{C}$, $\mathcal{D}$ be two small categories. Let $f\: : \: \mathcal{C}\to \mathcal{D}$ be a functor. Then it induces a functor
$$
f^{*}\: : \: sPsh(\mathcal{D})\to sPsh(\mathcal{C})
$$
...

**0**

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**1**answer

78 views

### Canonical colimit and cartesian product of simplicial sets

Let $K$ be a simplicial set and let $\Delta K$ be the category of simplices, i.e the category where the objects are simplicial maps
$$
\Delta[n]\to K
$$
and the maps $\phi\: : \: (\Delta[n]\to K)\to ...

**12**

votes

**1**answer

402 views

### Lemma 2.1.1.4 in Lurie's HTT

I have encountered a problem in understanding Lurie's proof of the following fact:
"Given a left fibration between simplicial sets $q:X \to S$, there exists a functor $$ho(S) \to Ho(sSet)$$ which is ...

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votes

**4**answers

796 views

### Topological Grothendieck Construction

Let $C$ be a small category and $F\colon C^{op}\rightarrow Set$ a functor. The Grothendieck construction is the category $F\wr C$ with objects being pairs $(c,x)$ where $c$ is a object of $C$ and ...

**8**

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**0**answers

352 views

### Replacing functors by topologically or simplicially enriched functors

I believe it is true that if a functor $F:Top\to Top$ preserves weak homotopy equivalences then it is related by a natural weak equivalence to some other such functor that is in some sense continuous, ...

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37 views

### contiguity and infinite complexes

Two simplicial maps $f: K^{(n)}\to L$, $g: K\to L$ are called contiguous if for every simplex $\tau\in K^{(n)}$ and it carrier $\sigma$ we have that $f(\tau)\cup g(\sigma)$ is a simplex in $L$.
It ...

**0**

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**0**answers

67 views

### Subdivision of a small category

I am reading about subdivision of a category from this paper:
http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Delgado.pdf
At the page 4, the author of this paper gives the first example ...

**10**

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**1**answer

305 views

### Tensor product of dendroidal sets: counter-examples

For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$.
...

**4**

votes

**1**answer

155 views

### fibrant generation of $sSet_{Quillen}$?

I wonder if someone has proved that $sSet_{Quillen}$ is not a fibrantly generated model structure ? Do we know something about the possible fibrant generation of $sSet_{Quillen}$ ?
Thanks

**1**

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**1**answer

88 views

### Cartesian products between cofibrant simplicial presheaves

Let $Psh(\mathcal{C})$ be the category of simplicial presheaves equipped with the projective model structure. The cartesian product between two representables presheaves is clearly again representable ...

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144 views

### A Cartesian model structure (and straightening for) on $n$-trivial simplicial sets

A pair $(X,tX)$, with $X$ a simplicial set and $tX$ a collection of simplices of $X$, is said to be stratified if no $0$-simplex is in $X$ and all degenerate simplices of $X$ are in $tX$. Recall a ...

**5**

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**1**answer

293 views

### Descent properties of spaces

I am trying to make sense of what is written in Rezk's draft http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf
In particular, I am referring to Proposition 2.3, which is there stated ...

**5**

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**1**answer

304 views

### Does homotopy invariance of homology follow from the structure of the simplex category $\Delta$?

Explicitly: Let $\Delta$ denote the simplex category, and $\mathscr{C}$ any small category, and fix a functor $F:\Delta \rightarrow \mathscr{C}$ such that $F\Delta^0$ is terminal. Also, assume ...

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votes

**2**answers

681 views

### A more natural proof of Dold-Kan?

The Dold-Kan correspondence gives an equivalence of categories between $SAb$, the category of simplicial abelian groups, and $Ch_{\geq 0}$, the category of non-negatively graded chain complexes of ...

**2**

votes

**1**answer

138 views

### recognising weak equivalences of simplicial sets

$\require{AMScd}$ I am interested in detecting using a lifting property when a map $f:X\to Y$ in $sSet$ (with the standard Kan model structure) is a weak equivalence. In the paper Weak Equivalences ...

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**0**answers

112 views

### Equalizer in Free groups

Let $F_n$ be the free group generated by $x_1,\cdots,x_n$ and for each $1\leq i\leq n$, let a homomorphism $d_i:F_n\to F_{n-1}$ be defined as follows:
$d_i(x_r)=x_r$, if $i>r$;
$d_i(x_r)=1$, if ...

**1**

vote

**1**answer

131 views

### Does the nerve functor preserve fibrations?

As asked in the title but more specifically: does the nerve functor from Cat to sSet map a fibration between groupoids to a Kan fibration ?
By fibration of groupoids I mean a fibration for the ...

**2**

votes

**1**answer

156 views

### Is the discrete nerve of a small category a complete Segal space?

While reading Rezk's paper "A model for the homotopy theory of homotopy theory", I found a remark which contradicts a guess of mine, but I can't see where I am wrong (perhaps it might be a silly ...

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**1**answer

100 views

### contracting homotopy on simplicial sets

Let $X$ be a topological space and let $PX$ be its space of paths. Let $I=[0,1]$ with coordinate $s$. There is an homotopy
$$
F\: : \: I\times PM\to PM
$$
Defined by $F(s,y)(t):=y(st)$. This map is an ...

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**1**answer

206 views

### To what extent are homotopy colimits over a weakly contractible category “determined by local data”?

[I'd be very happy for a better question title, if anyone has any suggestions.]
I have a category $C$, two functors $F,G : C \to \mbox{Cat}$, a natural transformation $\alpha : F \to G$, and a ...