The simplicial-stuff tag has no wiki summary.

**7**

votes

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558 views

### When is the projective model structure cartesian? When is the internal hom invariant?

If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category $Fun(D,M)$ where the weak equivalences are the ...

**11**

votes

**1**answer

317 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 ...

**9**

votes

**0**answers

128 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

172 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**

votes

**1**answer

145 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**

votes

**1**answer

259 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 ...

**4**

votes

**0**answers

136 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, ...

**1**

vote

**0**answers

54 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 ...

**10**

votes

**3**answers

940 views

### What are the fibrant objects in the injective model structure?

If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial ...

**3**

votes

**1**answer

230 views

### Which 2-coskeletal simplicial sets is the nerve of a category?

Let ${\mathrm{tr}}_2$ be the truncation functor that takes a simplicial set and restricts it to dimensions at most 2. Its right adjoint is the 2-coskeleton functor. NLab says that the nerve of a small ...

**1**

vote

**0**answers

68 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**

votes

**1**answer

245 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 ...

**0**

votes

**3**answers

80 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

85 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 ...

**2**

votes

**2**answers

346 views

### When do zero-simplices of a simplicial diagram determine its homotopy colimit?

Suppose that I have a diagram of simplicial sets $X_\bullet:\mathscr{C} \to Set^{\Delta^{op}},$ with $\mathscr{C}$ a small category such that for each $C \in \mathscr{C},$ $X_\bullet(C)$ is a Kan ...

**5**

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**0**answers

103 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 ...

**4**

votes

**2**answers

292 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 ...

**4**

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**0**answers

245 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 ...

**6**

votes

**1**answer

159 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 ...

**4**

votes

**1**answer

273 views

### Generalizing the circle packing theorem to 3-dimensions

The circle packing theorem (Koebe–Andreev–Thurston theorem) states that every finite planar graph is the nerve of some disk packing in the plane, where the nerve of a packing $P$ is a graph $G=(V,E)$, ...

**1**

vote

**1**answer

113 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}\ ...

**4**

votes

**1**answer

397 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 ...

**0**

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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 ...

**1**

vote

**1**answer

93 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

74 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 ...

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**0**answers

178 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 ...

**4**

votes

**1**answer

196 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 ...

**12**

votes

**2**answers

492 views

### Exponentiation in finite simplicial sets

A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. My question is: if $A$ and $B$ are finite simplicial sets, does this imply that the simplicial set ...

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votes

**6**answers

2k views

### Simplicial homotopy book suggestion for HTT computations

I'm struggling through Lurie's Higher Topos Theory, since it appears that someone reading through the book is expected to be somewhat comfortable with simplicial homotopy theory. The main trouble ...

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**0**answers

24 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 ...

**1**

vote

**1**answer

87 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**

votes

**1**answer

76 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 ...

**8**

votes

**4**answers

775 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**

votes

**0**answers

346 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, ...

**12**

votes

**1**answer

396 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

**0**answers

34 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 ...

**10**

votes

**1**answer

294 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

154 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

**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 ...

**5**

votes

**1**answer

289 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 ...

**1**

vote

**1**answer

83 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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**0**answers

142 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 ...

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**0**answers

251 views

### Left adjoints to several inclusions of homotopy simplicial model categories

The left adjoint to the inclusion $sGrp\hookrightarrow sPtSet$ of the category of simplicial groups into the category of pointed simplicial sets is homotopy equivalent to the loop suspension functor ...

**5**

votes

**1**answer

296 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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**1**answer

13k views

### If I want to study Jacob Lurie's books “Higher Topoi Theory”, “Derived AG”, what prerequisites should I have?

I've been told that it's important to know modern physics, Differential Geometry and Algebraic Topology for understanding higher structures. Is there any other prerequisite for understanding Lurie's ...

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votes

**2**answers

665 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

136 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 ...

**1**

vote

**1**answer

126 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 ...

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votes

**0**answers

111 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 ...

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votes

**4**answers

834 views

### Is there a sense in which the homotopy theory of simplicial sets is the “paradigmatic” one?

I could not come up with a better title for my question.
What I am asking is this (preemptive excuses to all experts in homotopy theory for naivetes of all kinds you may find herein):
the ...