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

7
votes
1answer
142 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 ...
4
votes
0answers
87 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,...
4
votes
1answer
103 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 ...
5
votes
0answers
56 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 \...
4
votes
0answers
82 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 (...
12
votes
1answer
252 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,...
6
votes
1answer
196 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 ...
3
votes
0answers
140 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\...
11
votes
0answers
149 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(...
2
votes
0answers
144 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 ...
3
votes
1answer
231 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 $n$, $...
7
votes
0answers
220 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 ...
4
votes
0answers
133 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. ...
7
votes
1answer
308 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 ...
5
votes
1answer
175 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 $\...
2
votes
0answers
56 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 ...
7
votes
1answer
225 views

Compatibility of Grothendieck construction with pullback

Suppose $D$ is an $\infty$-category, then we have the equivalence $$ \text{Fib} (D) \substack{ \text{St} \\ \longrightarrow \\ \cong \\ \longleftarrow \\ \text{Un}} [ D^\text{op}, \mathbf{Kan}]$$ ...
5
votes
1answer
149 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, ...
4
votes
1answer
201 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 $\text{lim }X_\bullet \to \text{holim } X_\bullet$...
6
votes
0answers
117 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 ...
1
vote
0answers
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....
13
votes
1answer
274 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 ...
5
votes
0answers
183 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 ...
4
votes
3answers
152 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 ...
8
votes
1answer
221 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 \...
5
votes
0answers
136 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 $ \...
4
votes
0answers
87 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 (...
4
votes
1answer
300 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 ...
12
votes
0answers
148 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 ...
5
votes
0answers
208 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 ...
12
votes
1answer
256 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 $\{...
7
votes
1answer
203 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_{...
4
votes
0answers
165 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 $\mathcal A$...
5
votes
0answers
62 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, ...
3
votes
0answers
222 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 ...
10
votes
1answer
490 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 ...
3
votes
0answers
91 views

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 ...
2
votes
0answers
157 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:...
7
votes
1answer
269 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 ...
8
votes
0answers
141 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 $4k$-...
4
votes
1answer
150 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 ...
7
votes
2answers
300 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 ...
3
votes
1answer
232 views

Homotopy equivalence of geometric realizations

This question is related with this one. For simplicial complex (which we have to assume is ordered as explained in the answer of the linked question) we have a construction of geometric realization ...
1
vote
1answer
210 views

A groupoid which is homotopy equivalent to $BG$

Let $G$ be a finitely generated group, then its action groupoid $BG$ is a simplicial set. In fact $BG$ is the nerve of a groupoid where the set of objects is given by a point $*$ and the set of maps ...
4
votes
1answer
231 views

Homology of simplicial complex versus homology of simplicial _set_

Let $K$ be a simplicial complex: it consists from the set (called the set of vertices) and a family of subsets of set of vertices satisfying the property of being closed under taking subsets (those ...
4
votes
0answers
164 views

category of simplicial filters

I am looking for references discussing the category $Filt$ of filters (in the sense of set theory, details below), its simplicial category $sFilt$ and its full subcategory $Top\hookrightarrow sFilt$ ...
10
votes
1answer
192 views

Realisation of maps between spheres by simplicial maps

Let $K^n_0 := \partial \Delta_{n+1}$ the simplicial set obtained by removing the $(n+1)$-simplex from the standard simplex. This gives a simplicial decomposition of the sphere $S^n$. More generally, ...
16
votes
2answers
572 views

Teaching Steenrod Operations

I am teaching a class on topology and want to introduce Steenrod Operations. I have talked about simplicial sets and classifying spaces of groups but have not talked about Eilenberg–MacLane spaces. ...
18
votes
4answers
1k views

An abstract nonsense proof of the Hurewicz theorem

The ordinary homotopy groups of a space $X$ are the homotopy groups of the corresponding singular simplicial set $Sing(X)$. The ordinary homology groups of $X$ are the homotopy groups of the ...
3
votes
0answers
113 views

$\pi_0$ in arbitrary category of simplicial objects

Let $\mathcal C$ be a category (let it be pointed and cocomplete) such that the category of simplicial objects $s\mathcal C$ is a model category. In particular, I'm interested in two cases: $\mathcal ...