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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20 views

### Maps of simplicial schemes

Let $X$ be a $k$-scheme and $G$ a finite group. We may view $X$ as a simplicial scheme. What is the definition of a $G$-torsor $P\to X$ in the category of simplicial schemes?
Let $BG$ be the usual ...

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37 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

310 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

298 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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73 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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197 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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204 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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**1**answer

152 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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**1**answer

155 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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92 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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**1**answer

108 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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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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83 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

267 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

206 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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144 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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164 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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**1**answer

225 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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**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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223 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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275 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

312 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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186 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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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

230 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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158 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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**1**answer

207 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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122 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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32 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

281 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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189 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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156 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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240 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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143 views

### 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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92 views

### 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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**1**answer

303 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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158 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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218 views

### 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

271 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

207 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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168 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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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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228 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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501 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 ...

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### Additional structure on augmented simplicial sets

I am currently studying augmented simplicial sets with some additional degeneracies. I was wondering if it is a structure that was already identified somewhere. This additional data is a family of ...

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159 views

### Any two faces of the standard simplex are “homotopic”?

It's intuitively clear that any two faces of the standard simplex are homotopic, more formally, for any $0\le i<j\le n$, there should be a (surjective) map in the category of simplicial sets,
$h:...

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272 views

### Is every category equivalent to the fundamental category of a directed space?

I was wondering if there was some work in directed algebraic topology to define a classifying directed space of a category? By that I mean the following.
In (undirected) algebraic topology, we ...

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146 views

### Local formula for the signature of $4k$-manifold

In "The Euler characteristic is the unique locally determined homotopy invariant of finite complexes" on page 61 in the penultimate paragraph, Levitt mentions that if one restricts to compact PL $4k$-...

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**1**answer

157 views

### Reedy cofibrancy of the bar construction for algebras

Let $O$ be a $C$-colored operad taking values in a model category $M$ (that may very well have more nice properties if needed; think for the moment of (bounded) cochain complexes over a field). We ...

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358 views

### Classifying space as the geometric realization of the nerve of $G$ viewed as a small category

Let $C$ be a small category: we define its nerve $(N(C)_k)_k$ as the following simplicial set: $N(C)_0=Ob(C)$ (the set of objects), $N(C)_1=Mor(C)$ (the set of all morphisms) and $N(C)_k$ to be a set ...