# 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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### Dugger and Spivak's combinatorial proof of Joyal's isofibration theorem, a fortiori?

In what follows, let $E^n = \operatorname{cosk}_0(\Delta^n)$
Joyal's isofibration theorem says precisely
An inner fibration $p:X\to Y$ of quasi-categories is a fibration for the Joyal model ...

**6**

votes

**2**answers

568 views

### Theorem 2.1.2.2 Higher Topos Theory

At the page 74 of HTT, there is the following theorem
Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....

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**0**answers

146 views

### Is the Thomason model structure on Cat simplicial? Is it a monoidal model category?

The Thomason model structure on the category of small categories is transferred from the Quillen model structure on simplicial sets along the right adjoint $Ex^2 \circ N$ (where $N$ is the nerve), i.e....

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132 views

### Higher homotopical information in racks and quandles

A quandle is defined to be a set $Q$ with two binary operations $\star,\bar\star\colon\ Q\times Q\to Q$ for which the following axioms hold.
Q1. a $\star$ a = a
Q2. (a $\star$ b) $\bar\star$ b = (a $...

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**1**answer

347 views

### Is there an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology?

There are numerous expositions of simplicial homology in the literature.
Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes.
Hatcher in “Algebraic ...

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**1**answer

204 views

### Simplicial set are to cubical sets what simplicial complexes are to …?

Simplicial sets and cubical sets (with or without connections) are defined as presheaves over some indexing categories. There is a full subcategory of simplicial sets that we can identify with the ...

**3**

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**1**answer

127 views

### Cellularity of anodyne extensions?

Are the anodyne extensions of simplicial sets always relative cell complexes of horn inclusions? (i.e. there is no need to consider retracts)
If not, is there a known counterexample?
Similarly, does ...

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**0**answers

45 views

### Localization at the left edge of a coherent left horn inclusion

Let $\Lambda^n_0 \hookrightarrow \Delta^n$ be a left horn inclusion for $n>1$. Then consider $\mathfrak{C}(\Lambda^n_0)\hookrightarrow \mathfrak{C}(\Delta^n)$, and we have a cofibration $[1]_{\...

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**1**answer

338 views

### $BG$ the stack, $BG$ the simplicial presheaf

I have a theoretical question about comparing two objects that I have recently come across.
For concreteness, let us work over the category $C$ of schemes over $k$. Let $G$ be an algebraic group over ...

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**2**answers

312 views

### Do finite simplicial sets jointly detect isomorphisms in the homotopy category? [duplicate]

Let $\mathcal{H}$ denote the homotopy category associated with the Kan-Quillen model structure on $\mathbf{sSet}$. Suppose we have a map $f\colon X \to Y$ between Kan complexes, such that for every ...

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**0**answers

77 views

### Axiomatization of the shuffle decomposition

I am trying to figure out an axiomatization for Reedy categories such that the product of representables admits a shuffle decomposition. Cisinski suggested imposing a condition, which we will state ...

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204 views

### Homotopy pullbacks of simplicial sets; Joyal vs Kan-Quillen model structures

I am interested in comparison of homotopy pullback squares in the category of simplicial sets with respect to Joyal' model structure and Quillen's one.
Suppose we are given a (strict) pullback square
...

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**0**answers

240 views

### Is there a citeable source for generators and relations of simplicial sets?

Simplicial sets can be specified using generators and relations,
in complete analogy with groups, rings, etc.
More precisely, a system of generators of relations for a simplicial set
consists of a ...

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votes

**1**answer

155 views

### Direct comparison from the Rezk hom to the hom of a simplicial category along the coherent nerve?

Consider the following construction: Define $G_n$ to be the contractible groupoid on $n+1$ objects. Choosing a linear order on the objects of each $G_n$ turns $G_*$ into a cosimplicial object. ...

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votes

**1**answer

158 views

### Simplicial nerve functor commutes with opposites

There are two "opposite" functors:
$$ op_\Delta\colon sSet\to sSet$$
and
$$op_s\colon sCat\to sCat.$$
The first takes a simplicial set to its opposite simplicial set by precomposing with the opposite ...

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**0**answers

93 views

### Simplicial homotopy groups - reference request

I am looking for a reference for the definition 2.6 in https://ncatlab.org/nlab/show/simplicial+homotopy+group, which states
"The simplicial homotopy groups of any simplicial set, not necessarily Kan,...

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votes

**1**answer

110 views

### $X^K$ a Kan complex, without model structure or anodyne extensions

If $X$ is a Kan complex, then it is an easy consequence of the existence of the Quillen model structure or of the basic theory of anodyne extensions, that $X^{K}$ is also Kan. However, I am interested ...

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votes

**0**answers

64 views

### 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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votes

**0**answers

89 views

### Minimal infinity categories in the Segal space picture

There is a well-known notion of a minimal Kan complex (see Goerss/Jardine's book) which is generalized to a minimal quasi-category in these notes by Joyal
www.math.uchicago.edu/~may/IMA/Joyal.pdf
(...

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**1**answer

276 views

### What is the coskeleton tower of a quasi-category?

I was giving a talk in a seminar, and I mistakenly said that the coskeleton tower of a quasi-category was its Postnikov tower. Someone corrected me, but a discussion then ensued about what, precisely,...

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**1**answer

207 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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**0**answers

145 views

### Comparing two simplicial resolution in a model category

Suppose that is a nice model category $\mathbf{M}$ (cofibrantly generated,...).
Suppose we have a two diagrams
$$F,G: \Delta^{op}\rightarrow \mathbf{M} $$
and a natural transformation $\nu: F\...

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175 views

### Are the Alexander-Whitney and Eilenberg-Zilber maps homotopy inverse in arbitrary abelian categories?

Let $\mathcal{A}$ be a monoidal abelian category. Let $A$ and $B$ be simplicial objects in $\mathcal{A}$, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(...

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votes

**1**answer

233 views

### On the paradox that $n$-coskeletal simplicial sets model all homotopy types

Please help me resolve the following paradox:
False claim: Let $X$ be an $n$-coskeletal, $n$-connected simplicial set. Then $X$ is weakly contractible.
Actually, I suppose the claim is ...

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votes

**1**answer

235 views

### Bisimplicial sets and homology

I'm not sure about the following result:
Theorem (?): Let $f_{\bullet,\bullet}: X_{\bullet,\bullet}\rightarrow Y_{\bullet,\bullet}$ a map of bisimplicial sets such that for any
natural number $n$, $...

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votes

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224 views

### Does geometric realization commute with passing to the compactly generated topology?

My question is in the title, but here is a more detailed formulation:
Let Top be the category of all topological spaces and continuous maps, and let CGTop be the subcategory of compactly generated ...

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votes

**0**answers

276 views

### Homotopy equivalences between singular simplicial sets

Let $A$ be a topological space, and $S(A)$ its singular simplicial set. If $|S(A)|$ denotes the geometric realisation, it is known that the co-unit $\epsilon_A\colon|S(A)|\to A$ is a weak-equivalence. ...

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**1**answer

318 views

### About fibrations with fibre Eilenberg-MacLane spaces

Let $f: E\rightarrow B$ be a Kan fibration between pointed connected Kan complexes with fibre the Eilenberg-MacLane space $\mathrm{K}(M, n), n\geq 2, M$ an abelian group. Assume $f$ induces an ...

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**1**answer

205 views

### 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 $\...

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**0**answers

58 views

### Transfer map between simplicial manifolds

Let $M^m$ and $N^n$ be two triangulated oriented and closed manifolds and $f:M\to N$ a simplicial map. For each $a\in H_p(N)$ we may consider its homological transfer $f_!a\in H_{m-n+p}(M)$.
I want ...

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**1**answer

232 views

### Compatibility of Grothendieck construction with pullback

Suppose $D$ is an $\infty$-category, then we have the equivalence
$$ \text{Fib} (D) \substack{ \text{St} \\ \longrightarrow \\ \cong \\ \longleftarrow \\ \text{Un}} [ D^\text{op}, \mathbf{Kan}]$$
...

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votes

**1**answer

161 views

### Reference request for statement on nlab: Reedy (co)fibrancy of (co)simplicial objects

On the nlab
http://ncatlab.org/nlab/show/Reedy+model+structure#fibrant_and_cofibrant_objects
it is claimed that a simplicial object in a model category, in which all monomorphisms are cofibrations, ...

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votes

**1**answer

215 views

### Homotopy limits of Simplicial sets

Consider a small diagram $X_\bullet$ of simplicial sets. Under what sufficient conditions (apart from $X_\bullet$ being injective fibrant), is the map $\text{lim }X_\bullet \to \text{holim } X_\bullet$...

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123 views

### Dowker and neighborhood complexes: reference wanted

Let $R$ be a 0-1 matrix whose rows or columns are maximal.
Q1. Is there a name for such a matrix (or, e.g., a corresponding relation)?
From 0-1 matrix corresponding to an abstract simplicial ...

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vote

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33 views

### Criterion for bisemisimplicial sets to be a manifold

It is well known that a finite simplicial complex is a manifold of dimension $n$ iff the the link of each vertex is homeomorphic to $\mathbb{S}^{n-1}$.
Are there any criteria for weaker structures? E....

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**1**answer

286 views

### Weak complicial sets: Are the morphisms too strict?

In Verity's first paper on weak complicial sets, he shows that every strict complicial set is a weak complicial set. He also showed in an earlier paper that the full subcategory of stratified ...

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191 views

### What good is a cofibration of categories?

I'm working on a problem where I have a cofibration of $V$-enriched categories $f: A\to B$, and would like to study the induced functor on presheaves $[B,V] \to [A,V]$. We can assume $V$ is a ...

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votes

**3**answers

160 views

### 0-1 matrix corresponding to an abstract simplicial complex

Let $A$ be a 0-1 matrix whose columns are maximal. We can associate its rows with vertices and columns with simplices in an abstract simplicial complex. Conversely, given an ASC, we can encode it in a ...

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votes

**1**answer

264 views

### Is totalization (of a cosimplicial category) a part of some adjunction?

For a diagram category $\Gamma$ and and a cocomplete category $\mathcal{C}$, we have an equivalence $$\mathrm{Fun}(\Gamma,\mathcal{C}) \simeq \mathrm{Adj}(Set^{\Gamma^{Op}},C)$$ where for $F: \Gamma \...

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### Resolution of Simplicial Commutative Rings

I have just started learning some derived algebraic geometry. I was told that (if $ \mathrm{char}(\mathbb{K})=0 $) using commutative differential graded algebras in negative degree (for short $ \...

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### Does a fibrant simplicial set give fibrant diagram

If $Y$ is a fibrant simplicial set and $\Delta^{\bullet}$ is the cosimplicial simplicial set, is $Y(\Delta^{\bullet})$ (i.e. $n^{th}$ simplicial set is $n \mapsto Y(\Delta^{n})=Hom(\Delta^{n},Y)$) a (...

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votes

**1**answer

305 views

### Kan condition for bar construction

Let $T$ be a monad on a concrete category $\mathcal{C}$, and $A$ an algebra over $T$. The bar construction is a simplicial object in the category $\mathcal{C}^T$ of algebras which we can think of a ...

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166 views

### Kan's simplicial formula for the Whitehead product

In his article on Simplicial Homotopy Theory (Advances in Math., 6, (1971), 107 –209) Curtis quotes a formula (on page 197) for the Whitehead and Samelson products in a simplicial group $G$. The ...

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### relative spectrum in derived algebraic geometry

I am trying to understand how much it is possible to extend the notion of spectrum of a qcoh sheaf of algebras to stacks.
More precisely, given a scheme $S$ and a stack $F$ of cohomological cdga's ...

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**1**answer

287 views

### Configuration spaces, Ran spaces, free semilattices, Vietoris spaces and power objects

These are five important constructions and I would like to know how they are related.
The $n$th unordered configuration space of a space $X$ is
$$
\operatorname{UConf}_n(X):=\{\text{embeddings of $\{...

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**1**answer

210 views

### Homotopy type of the semi-simplicial set of symmetric groups

Consider the collection of symmetric groups $\{\Sigma_n\}_{n\geq1}$ as a semi-simplicial set (i.e. a simplicial set without degeneracies) as follows. Consider $i\in\{1,\dots,n+1\}$ and $\pi\in\Sigma_{...

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**0**answers

171 views

### Long exact sequence of Quillen derived functors

Let $F:\mathcal A\to \mathcal B$ be an additive right exact functor between abelian categories $\mathcal A$ and $\mathcal B$. Then for a short exact sequence
$$
0\to A\to B\to C\to 0
$$
in $\mathcal A$...

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votes

**0**answers

66 views

### Is a retract of contractible augmented simplicial complex contractible?

Suppose the augmented simplicial complex $X$ is a retract of the augmented simplicial complex $Y$ and that $Y$ is contractible. Is $X$ contractible? There is an obvious candidate for a contraction, ...

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236 views

### Homotopy limits commute with right Quillen functors

In particular I'm interested in this situation. Let $sSet^{2}$ denote the category of bisimplicial sets with diagonal model structure (weak equivalences are diagonal weak equivalences and cofibrations ...

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**1**answer

508 views

### On HTT's Lemma 3.3.4.1

While studying the book Higher Topos Theory I have encountered some difficulty with Lemma 3.3.4.1, which says that the pullback along a cartesian fibration of a map q such that $q^{op}$ is cofinal is ...