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1
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0answers
115 views

Certain (pre) simplicial sets have torsion-free homology?

Suppose $X$ is a pre-simplicial set defined on a finite vertex set $\{v_1, v_2, \ldots, v_k\}$ recursively as follows: Let $X_1$ have a single vertex $\{v_1\}$. For $i > 1$, $X_i$ is obtained ...
8
votes
0answers
412 views

Does the category of strict $2$-categories together with Dwyer-Kan equivalences provide a model for $(\infty,1)$-categories?

The question is the title. In what follows, all $2$-categories and $2$-functors will be strict. Let $2-Cat$ denote the categories whose objects are $2$-categories and whose morphisms are ...
3
votes
2answers
554 views

Examples of Simplicial Groupoids in Nature

For me, a simplicial groupoid is a simplicial object in ${\mathbf{Grpd}}$. I am more general than Goerss-Jardine in this definition. Do you have examples simplicial groupoids that occur in nature? ...
0
votes
1answer
289 views

Is there a dual notion for the Nerve functor?

Say in the most classical case, we probe a topological space $X$ by the n-simplices $\Delta^n$ by using the nerve functor $Hom_{Top}(-,X)$. Is another functor $Hom_{Top}(X,-)$ of any use, or is there ...
1
vote
1answer
243 views

The boundary map of Kan simplicial sets

I don't know whether or not the following is a research level question, but since it is concerned with simplicial sets and since they are very popular these days, I think this is the right place to ...
2
votes
1answer
249 views

Is the nerve of a category a fully faithfully functor up to homotopy?

Let $F,G:C\to D$ be naturally isomorphic functors. Taking the nerve, is $NF,NG:NC\to ND$ homotopy equivalent? Conversely, given a simplicial map $f:NC\to ND$, does there exists a functor $F:C\to D$ ...
1
vote
1answer
220 views

Is the nerve of a groupoid an EM space?

Let $G$ be a connected groupoid. Is the nerve $BG$ a $K(\pi, 1)$, and if so, is there a groupoid homomorphism $f:G\to \pi$ that induces the homotopy equivalence?
4
votes
0answers
189 views

Is the class of inner-anodyne morphisms right-cancellative with respect to the of the class of monomorphisms?

Recall: Given a category $A$, and two classes of morphisms $S,S'$, we say that $S$ is right-cancellative with respect to $S'$ if for any pair of maps $f\in S, g\in S'$ such that $gf$ is defined, we ...
1
vote
0answers
187 views

Proof for a simplicial morphism

Sorry for the title but I can't come up with another one... ... Suppose X and Y are simplicial sets and $f : X \rightarrow Y$ is a map such that: 1.) f maps dimensions the right way, that is ...
11
votes
2answers
1k views

What is a symmetric monoidal $(\infty,n)$-category?

This question arose from reading Jacob Lurie's "Classification of topological field theories" paper. In that paper, he uses complete $n$-fold Segal spaces as a model for $(\infty,n)$-categories, but ...
5
votes
0answers
199 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 ...
7
votes
2answers
599 views

Do simplicial objects in a Topos form a model category?

Sometimes people say "If you don't like the word 'topos', just think the category of Sets", but I'm not sure to what extent this analogy holds. The real question here is, do simplicial object in a ...
2
votes
0answers
185 views

The Mayer-Viertoris exact sequence as a (Zariski) descent spectral sequence.

For certain 'spaces' $U,V$ (they are certain Henselizations of subvarieties) I would like to compute (certain etale) cohomology of $U\cup V$ in terms of the corresponding cohomology of the diagram ...
1
vote
4answers
481 views

Decomposition of inner horns

Hi guys, I have a question. Prove or disproof the statement: Any inner horn $\Lambda[n,k],0< k< n $ admits a filtration $\mathbf{n}<\cdots<\Lambda[n,k]$, such that each step is filling an ...
9
votes
2answers
1k views

Semi-simplicial versus simplicial sets (and simplicial categories)

Hi, Let's denote by "semi-simplicial set" a simplicial set without degeneracies, i.e. a contravariant functor from the category $\Delta_{inj}$ of finite linearly ordered sets and order preserving ...
21
votes
1answer
10k 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 ...
10
votes
1answer
385 views

The weak equivalences in the covariant model structure

Let $S$ be a simplicial set. Recall that there is a model structure, called the covariant model structure (see HTT ch. 2 and this question), on $\mathbf{SSet}/S$ such that: The cofibrations are the ...
6
votes
1answer
485 views

What does the “category” of $(\infty,1)$ category look like.

One knows that in higher category theory, the category of $(\infty,n-1)$ categories is naturally an $(\infty,n)$ category ,(I use the word category to mean category in the correct weakened sense). ...
4
votes
1answer
333 views

A “join” of ω-categorical simplices

Recall that the category of level trees $\mathcal{T}$ is defined to be the category $[\mathbb{N}^{op},\Delta_a]$, where $\Delta_a$ is the skeleton of the category of finite possibly empty linearly ...
13
votes
1answer
419 views

When can a contractible 2-complex be embedded in R^3?

Let $X$ be a contractible 2-dimensional simplicial complex. Are there nice necessary and sufficient conditions for $X$ to be embeddable in $\mathbb R^3$? Clearly it is necessary that the link of ...
10
votes
1answer
800 views

The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?

Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying ...
6
votes
2answers
345 views

$N$-step simplicial complexes

Recently, answering a question here, Dror Bar-Natan observed that «way too often two-step complexes have a natural extension to become many-step complexes». By such a thing I mean (and I think Dror ...
1
vote
1answer
156 views

Is there a linear embedding of a simplical 3-complex in R^6?

I've heard that there always is an embedding in $R^7$ (can someone provide a reference for that?) and this number cannot be lowered in general. But I'm interested in a somewhat special case, namely: ...
1
vote
2answers
540 views

Nerves of simplicial objects in categories/Waldhausen's S-construction

Is there a good nerve-like functor from simplicial objects in categories to simplicial sets which takes level-wise equivalences of categories to weak equivalences? To give this some context, I'd ...
9
votes
1answer
796 views

Is the simplicial completion of a localizer always a bousfield localization of the injective model structure?

Background Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following ...
4
votes
0answers
105 views

What is known about the number of permissible simplicial complexes given the number of k-cells?

Motivation: I am working on a problem that reduces to finding simplicial complexes given some data (details are unnecessary), but all I have managed to wrangle from my input is the number of cells of ...
8
votes
2answers
717 views

Gluing of manifolds and the Hausdorff condition.

Hi! I apologize in advance if this question is better fit for http://math.stackexchange.com/. Out of curiosity I'm interested in a particular case of the problem of what properties of a manifold is ...
2
votes
1answer
224 views

Induced pretopologies on sSet

Recall that the geometric realisation functor $| - |: sSet \to Top$ preserves products (choosing $Top = k Space$ or similar). Thus any given singleton Grothendieck pretopology on $Top$ gives rise to a ...
1
vote
1answer
290 views

Do bistellar flips preserve shellability?

I notice there is a strong connection between shellability of simplicial complexes and bistellar flips on these complexes; in particular, adding in a new facet of a shelling induces a bistellar flip ...
11
votes
2answers
1k views

What is the homotopy theory of categories?

I've heard that Grothendieck, in his letter "Pursuing Stacks," wanted to find alternative models for the classical homotopy category of CW complexes and continuous maps (up to homotopy), and one of ...
2
votes
0answers
293 views

simplicial deRham complex and model category structure

To every simplicial manifold is associated its simplicial deRham complex. Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) ...
0
votes
2answers
446 views

Collinear vertices and definition of k-simplex

On page 120 of his Basic Topology, Armstrong defines the $k$-simplex in $\mathbb{E}^n$ with verices $v_0,\ldots,v_k$ to be complex hull of said vertices. (A similar definition is given on Wikipedia). ...
4
votes
2answers
307 views

Simplicial presheaves that are colimits of themselves?

Suppose $C$ is a small category and $X_{\bullet}$ is a simplicial object in $C$. In particular, by composing with Yoneda $$y:C \to Set^{C^{op}}$$ $y(X)_{\bullet}$ is a simplicial presheaf. I believe ...
3
votes
1answer
477 views

Question on the interpretation of a presheaf category as a co-completion

The category of presheaves $Pre(C)$ on a small category $C$ is the category of functors $C^{op}\to Sets$. Since the category of sets is co-complete and every presheaf is a colimit of representable ...
36
votes
5answers
5k views

Is there a high-concept explanation for why “simplicial” leads to “homotopy-theoretic”?

My (limited) understanding is that simplicial methods tend to be used whenever you want some kind of nontrivial homotopy theory -- for instance, to get a nice model structure, you use simplicial sets ...
11
votes
4answers
822 views

Degeneracies for semi-simplicial Kan complexes

By a semi-simplicial set I mean a simplicial set without degeneracies. In such a thing we can define horns as usual, and thereby "semi-simplicial Kan complexes" which have a filler for every horn. ...
16
votes
1answer
1k views

Analogue of simplicial sets

This question is prompted by this one (and some of the comments that it drew). Simplicial complex is to ordered simplicial complex as $X$ is to simplicial set. The question is about $X$. Let ...
17
votes
2answers
724 views

Is there a discrete Cerf theory?

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the ...
7
votes
2answers
543 views

Historical Question about Simplicial Sets

I have a pretty easy historical question about simplicial sets. Unless I am mistaken, simplicial sets first came out of topology, explicitly from combinatorial topology and the study of simplicial ...
7
votes
1answer
356 views

delooping under Dold-Kan and simplicial delooping

What maps of simplicial sets exist between the image under the Dold-Kan correspondence of a chain complex shifted up in degree and the image under the right adjoint to simplicial looping of the ...
4
votes
1answer
239 views

Artin-Mazur codiagonal preserves Kan objects?

The Artin-Mazur codiagonal $\nabla:ssSet \to sSet$ is right adjoint to the total decalage functor $Dec:sSet \to ssSet$. The total decalage functor is defined to be precomposition with the ordinal sum ...
3
votes
2answers
492 views

Inner hom and geometric realization.

I would like to prove the following fact, which I learned from a previous MO question. Let $S_\cdot,T_\cdot\in\mathbf{sSET}$ be simplicial sets, and assume that $T_\cdot$ is Kan. Then there is a ...
8
votes
1answer
455 views

Nerve: Groupoids-> Kan Complexes. Nerve: Bicategories w. adjoints -> ?

If you take the nerve of a groupoid, you get a Kan complex. Question: Take a bicategory that has adjoints for 1-morphisms, which is one notion of 'weak' groupoid (if all 2-morphisms are ...
3
votes
0answers
136 views

Is the homotopy of a primitively generated Hopf algebra still primitively generated?

Let $A=\oplus A_n$ be a primitively generated graded Hopf algebra, where each $A_n$ is a simplicial group. This allows us to define the homotopy group $\pi_*(A)$. Question: is the graded Hopf algebra ...
2
votes
2answers
331 views

Algorithm that decreases the size of the simplicial triangulation

Hello! Let X be a topological space. We are considering only abstract simplicial complexes, i.e. a finite list of finite lists of integers. Is there any algorithm (more or less efficient?), that ...
8
votes
1answer
378 views

Does this “flipping lexicographic” ordering have a standard name?

I’ve run into the following straightforward variant of lexicographic ordering, and am wondering if it has a standard name. I’ve been calling it the flipping lexicographic ordering, for evident ...
8
votes
0answers
297 views

A combinatorial approximation functor sSet->qCat

Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap ...
3
votes
2answers
336 views

Model categories of simplicial objects

If $\mathcal{C}$ is a category, then surely the category of simplicial objects $s\mathcal{C}$ is not automatically a model category. What conditions must $\mathcal{C}$ satisfy in order for ...
11
votes
7answers
2k views

History of classifying spaces

Where did the idea and formal definition of the "classifying space of a (small) category" first appear? Added: As Andy Putman noted below, the "classical" early reference for this is G. Segal's ...
4
votes
0answers
58 views

maps of oo-stacks inducing monomorphisms on all homotopy sheaves

What is known about factoring morphisms of simplicial (pre)sheaves/$\infty$-sheaves through maps that induce monomorphisms on all homotopy sheaves? Is there any useful theory for that?