For questions about simplicial sets (functors from simplex category to sets), simplicial (co)algebras and simplicial objects in other categories; geometric realization, model structures, Dold-Kahn correspondence etc. Please do not use for questions about geometry of simplices nor about ...

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7
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95 views

History of the term “anodyne” in homotopy theory

There is a notion of an anodyne morphism (usually of simplicial sets). This type of morphisms is used, for instance, to establish basic properties of quasicategories. But the name is quite mysterious. ...
7
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74 views

Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
7
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103 views

A theorem of Kan regarding fibrant replacement

Recall that there is an adjunction $$\mathrm{Sd} \dashv \mathrm{Ex} : \mathbf{sSet} \to \mathbf{sSet}$$ where $\mathrm{Sd} (\Delta^n)$ is the first barycentric subdivision of $\Delta^n$. There is a ...
6
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0answers
151 views

Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$

I was wondering if I am in the right track here. Let $A := k[x,y]/(x^2 - y^3)$, the cuspidal curve. Obviously this isn't etale or smooth over $k$ so its cotangent complex is not contractible. Now, I ...
6
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157 views

Inverse functor in proof of Dold Kan Correspondence

I´m looking at the proof of the Dold-Kan correspondence. Let $SA$ be the category of simplicial objects in an abelian category $A$ and $CH_{\ge0}(A) $ the category of non-negative chain complexes in ...
5
votes
0answers
120 views

Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the simplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
4
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0answers
59 views

(Co)homology of propositional logic

Sorry if this is a rather vague question, but it seemed like something that might be interesting. Let $P$ be a family of propositions, and let $\mathcal L(P)$ be the set of all compound propositions ...
4
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65 views

Unique representation of a degenerate simplex

I recently learned about simplicial sets, and now I'm studying some basic properties. I've learned that every degenerate simplex is a degeneracy of a unique non-degenerate (nice) simplex. However, I ...
4
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0answers
67 views

Morita theory for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting? I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...
4
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0answers
143 views

Two definitions of equivariant sheaves

Let $G$ be a topological group. Here are two definitions of $G$-equivariant sheaves on a $G$-space $X$. (a) Define an $G$-equivariant sheaf by a sheaf $F$ (étalé space) equipped with a $G$-action ...
4
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207 views

Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
4
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115 views

What is a copresheaf on a “precategory”?

Let $\mathcal{S}$ be a category with pullbacks. A precategory $\mathbb{C}$ in the sense of Borceux and Janelidze [Galois Theories, §7.2] comprises three objects $C_0, C_1, C_2$ in $\mathcal{S}$ and ...
3
votes
0answers
14 views

Is the projection from the cyclic nerve to the simplicial nerve a cyclic map?

I’m trying to understand Loday’s proof of Theorem 7.3.11 in “Cyclic Homology” (2nd ed 1998). I have a problem right at the beginning where it says: The projection map $\text{proj}:\Gamma_\bullet G ...
3
votes
0answers
98 views

Associativity of the tensor product of dendroidal sets

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 $A$. ...
3
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0answers
38 views

Why aren't *weak* test categories enough?

Short version : What is the motivation behind the local condition in the definition of a test category ? Why do we want the slice categories $A/a$ to be weak test ? Long version : I start ...
3
votes
0answers
57 views

Converting a space to a simplicial set

Given a compact manifold and a Morse function, we obtain a convenient cellular decomposition. But suppose I have a (finite dimensional, possibly singular) space $X$ that is not a manifold and I wish ...
3
votes
0answers
54 views

An equality of some equalizers of simplicial sets.

$\newcommand{\cosimp}[1]{{#1}^\bullet} \newcommand{\sSet}{\mathsf{sSet}} \newcommand{\Set}{\mathsf{Set}}$ [I realize that the post is imposing.However, all the content of the problem is in the gray ...
3
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0answers
65 views

Does the $\mathcal{E}G$-construction arise form a comonad?

Let $G$ be a simplicial group and consider the adjunction $$ Fr:sSets\rightleftarrows G-sSets:U $$ where the left adjoint $Fr$ maps a simplicial set $X$ to the free $G$-simplicial set $G\times X$ and ...
3
votes
0answers
74 views

Is a simplicial subcomplex of a contractible simplicial CW-complex also contractible?

Let $Y,X$ be simplical CW-complexes (By that I mean functors $\Delta^{op}\to CW$. In other words, for each natural number $n$ there exist CW-complexes $X_n$ and $Y_n$ and there are face and degeneracy ...
3
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0answers
48 views

For which coverings by “geometrically nice” sets does the nerve admit “Vietoris-Rips-like” approximations?

It is well known that the nerve (or Čech complex) of a covering by metric balls is nicely approximated by the Vietoris-Rips complex. Being a flag complex on its 1-simplices, the latter is ...
3
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0answers
89 views

What is an “absolute, equational pushout”?

I was leafing through Joyal and Tierney's Notes on simplicial homotopy theory. In the first few lines of the section on the skeleton of a simplicial set they display the simplicial identities as a ...
2
votes
0answers
29 views

Kan's loop group construction

I'm looking for a good place to read about the loop group construction $G : \mathbf{sSet_0} \to \mathbf{sGrp}$ taking a reduced simplicial set $X$ and producing a simplicial group $GX$. I would also ...
2
votes
0answers
32 views

The identity map $\Delta^n \rightarrow \Delta^n$ is a basis for $H_n(\Delta^n, \delta\Delta^n; R)$

Show that the identity map $\Delta^n \rightarrow \Delta^n$ is a basis for $H_n(\Delta^n, \delta\Delta^n; R)$. Here $\Delta^n$ is the n-simplex, and I know $\delta \Delta^n$ denotes its ...
2
votes
0answers
78 views

Difference between two concepts of homotopy for simplicial maps?

I learn from Gelfand and Manin's Methods of Homological Algebra, Exercise 2 for I.4 that two maps $f,g\colon X\to Y$ between simplicial sets $X,Y$ are simply homotopic (maybe usually called simplicial ...
2
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0answers
29 views

When may the bar resolution be truncated for calculating $\operatorname{hocolim}$?

Suppose $D$ is a small category and $F\colon D\to sSets$ a functor. I want to calculate the simplicial set $$\operatorname{hocolim}F$$ via the bar construction: $\operatorname{hocolim}F$ is the ...
2
votes
0answers
74 views

Right Kan extension along a diagonal functor.

Denote $\newcommand{\simpcat}{\boldsymbol\Delta} \simpcat$ the simplex category (category of finite ordinals with non decreasing maps). The diagonal functor $\delta \colon \simpcat \to ...
2
votes
0answers
54 views

Carrier maps between simplicial complexes

Simplicial complexes are useful in proving things about distributed systems. We define and use simplicial maps between two such complexes, and these maps also seem to be standard objects of study in ...
2
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0answers
23 views

Explanation of notation for face and degeneracy

The (co)face are historically denoted by $d$, while the (co)degeneracy maps are denoted by $s$. Why is this, instead of the obvious $f$ and $d$? In the simplicial category $\Delta$, you also realize ...
2
votes
0answers
101 views

Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius

I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
2
votes
0answers
179 views

The Fundamental Group of a Polyhedron Depends Only on its $2$-Skeleton

Does anyone have a good quick proof of this using the Simplicial Approximation Theorem? I'm aware that it comes out as a corollary when considering edge paths and the edge group, but this seems like ...
2
votes
0answers
123 views

left inverse to trivial fibration is trivial cofibration

It is claimed that in the model category of simplicial sets (with usual model structure), a trivial fibration $X \to Y$ has a section, which is a trivial cofibration. Now, I see that there is a ...
2
votes
0answers
234 views

does it make sense to tensor a simplicial set with a simplicial ring?

Given a simplicial commutative ring $A$ and a simplicial set $K$ does $K \otimes A$ make sense as a (commutative) simplicial ring? I'm asking as I've seen the expression $S^n \otimes A$ written down ...
2
votes
0answers
94 views

Local injective model structure for simplicial presheaves

The category of simplicial presheaves on a small Grothendieck site $\mathcal{C}$ can be given a model structure by defining weak equivalences and cofibrations sectionwise. It's called the (global) ...
1
vote
0answers
23 views

Proof of Quillen's lemma about minimal fibrations

In Hovey's book $\textit{Model categories}$ (available online here http://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey--model-cats.pdf) one finds the classical result that any ...
1
vote
0answers
54 views

Topological insight on an algebraic theory.

I recently read about the marvelous bar construction. As far as I understand, it is something of a free resolution in an abstract setting. I'm wondering if it can be used to get topological insights ...
1
vote
0answers
15 views

Simplicial Complexes of Graphs notation question

I'm studying Jakob Jonsson's book Simplicial Complexes of Graphs very rigorously and in depth. I've been okay so far with the intensity and notation, but on page Chapter 3, section 2, page 30, I'm ...
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0answers
22 views

Criterion for projectively cofibrant diagrams?

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 category of functors $\mathbf{D}\to ...
1
vote
0answers
50 views

Mapping $X$ to $\text{sk}_2(X)$ while fixing $\text{sk}_1(X)$

UPDATE: The original question can be simplified to this: Given a finite simplicial complex $X$, can I find a continuous $f : X \rightarrow \text{skel}^2(X)$ that fixes $\text{skel}^1(X)$? Basically, ...
1
vote
0answers
43 views

Combinatorial definition of the homotopy groups of a quasi category?

The homotopy groups of a Kan complex are defined combinatorially, and they coincide with the homotopy groups of the geometric realization. For a general complex $X$, homotopy isn't an equivalence ...
1
vote
0answers
60 views

Geometric realization of function complexes of simplicial sets

Let $\operatorname{sSet}$ be the category of simplicial sets and $\operatorname{Top}$ the category of (say, compactly generated) topological spaces. There is a pair of adjoint functors called ...
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vote
0answers
53 views

A question about partitioning the unit cube into simplexes

Let $C$ be the unit cube in $\mathbb{R}^n$. I want to simplicize $C$ (i.e. partition it into simplexes), but I have to follow these two rules: I can't add any vertices to $C$. The vertices of each ...
1
vote
0answers
74 views

geometric realization of a free simplicial group

Call a simplicial group $G_{\bullet}$ free if for each $n$, the group $G_n$ is a free group. How does the geometric realization of $G_{\bullet}$ look like? Can its nondegenerate and ...
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vote
0answers
64 views

Can one define “simplicial” EM spaces?

Let $\Delta$ be the well-known simplicial category, and denote with $\widehat\Delta_\bf C$ the category of simplicial objects in $\bf C$ ($\widehat\Delta$ for short). Given $\mathcal ...
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70 views

Valuations, simplicial complexes and arithmetics

I am currently working on an Introduction to geometric probability (Klein, Rota, 1997). This book is very stimulating, and I find myself toying with the subject, and a note in particular (pp 95-97 for ...
0
votes
0answers
34 views

Is there a spectral sequence to estimate the connectivity of a homotopy limit of a cosimplicial space?

Given a diagram $$ D\colon \Delta\to sSet_* $$ i.e. a cosimplicial pointed simplicial set by $[k]\mapsto S^n\wedge D'([k])$ for some other diagram $D'\colon \Delta\to sSet_*$ and a fixed integer ...
0
votes
0answers
16 views

Are levelwise endofunctors of simplicial sets homotopical?

Let $Set$ be the category of sets and $sSet$ the category of simplicial sets (the category of functors from $\mathbb{\Delta}^{op}$ to $Set$). Every functor $F:Set \to Set$ induces a functor ...
0
votes
0answers
19 views

homotopy of simplicial maps between infinite complexes

I have such a problem. Let $K,L$ be simplicial complexes, where $K$ is connected, countable and locally finite. In particular we can express $$K=\bigcup_{i=0}^\infty K_i,$$ where $K_0$ is some full ...
0
votes
0answers
27 views

Restricting a continuous function between simplicial complexes to $2$-skeleton

I have a continuous function $f : X \rightarrow X$, where $X$ is an $8$-dimensional simplicial complex. I want to homotope $f|_{\text{skel}^2(X)} : \text{skel}^2(X) \rightarrow X$ to a function $g : ...
0
votes
0answers
45 views

An variation of “Lk” in simplicial complexes

Let $\mathcal S$ be a simplicial complex. For a simplex $S \in \mathcal S$, we have the closure, the star and the link $\operatorname{Cl} S := \{ F \in \mathcal S : F \leq S \}$ $\operatorname{St} S ...
0
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0answers
183 views

Generalizing Euler's Polyhedral Formula for Graph Embeddings in Higher Dimensions.

In the plane, Euler's Polyhedral formula tells us that $V - E + F = \chi$, where for graph embeddings we have that $\chi = 1$. Alternatively, we can think of a graph embedding as a simplicial ...