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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2
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1answer
35 views

Does the functor $\mathbf{cosk_n}:sSet\to sSet$ preserve Kan complexes?

On this nlab page a functor $\mathbf{cosk_n}:sSet\to sSet$ is constructed. Is $\mathbf{cosk_n}(K)$ of a Kan complex $K$ again a Kan complex?
5
votes
0answers
118 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 ...
3
votes
0answers
60 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 ...
3
votes
1answer
97 views

Cech cohomology and cohomology of a category : a cluster of questions.

I apologize in advance : what follows is a bit of a mess. Also, I think it might be a big tautology, but i don't see it yet. My question is about the rapport of Cech cohomology and cohomology of a ...
1
vote
1answer
40 views

Contractibility of topological spaces associated to posets

Suppose $\mathcal{P}$ is a partially ordered set. To $\mathcal{P}$ we can associate a simplicial complex $K(\mathcal{P})$ whose $n$-simplices are the chains of length $n+1$ in $\mathcal{P}$. Since ...
3
votes
1answer
51 views

Understanding Quillens Theorem A

Let me restate the theorem: Let $F\colon\mathcal{C}\to\mathcal{D}$ be a functor. If $F\downarrow x$ is contractible for every $x\in\operatorname{Ob}(\mathcal{D})$, then $F$ is a homotopy ...
0
votes
1answer
52 views

Orientation of a simplicial decomposition of $D^2\times \mathbb{S}^1$

This should be a simple problem but I started in on it and ran into something I don't understand. Essentially, I want a simplicial decomposition of $D^2\times \mathbb{S}^1$. $D^2$ is the basic ...
2
votes
1answer
137 views

augmented algebras and their morphisms

Let $R$ be a commutative unital ring and $A$ an associative (unital) $R$-algebra. What is an augmented $R$-algebra? A (unital) $R$-algebra $A$, together with a (unital) ring morphism $\varepsilon: ...
2
votes
3answers
173 views

Why are simplicial categories useful?

By simplicial category here I mean simplicially enriched category, i.e. all $Hom$-sets are simplicial sets and compositions are morphisms of simplicial sets. My question is the following. Suppose I ...
5
votes
1answer
148 views

Homology other than singular?

Usually, one defines $n$-th homology functor on topological spaces as the composite functor $$ \mathbf{Top} \to [\Delta^\mathrm{op},\mathbf{Set}] \to [\Delta^\mathrm{op},R\!-\!\mathbf{Mod}] \overset C ...
2
votes
1answer
77 views

Show that two different embeddings of the figure-eight in the torus are not homotopic

Note, we can express the torus $|X.| \cong T$ as a square with edges denoted by $e$ and $f$, the diagonal by $g$, and faces $T_1$ and $T_2$, and a single vertex $v$, with appropriate identifications. ...
0
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0answers
26 views

Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic) [closed]

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
7
votes
1answer
110 views

Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic)

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
1
vote
1answer
87 views

Show that any continuous map $f: X \rightarrow Y$ induces a map of semisimplicial sets $Sing(X). \rightarrow Sing(Y).$

I want to show that any continuous map $f: X \rightarrow Y$ induces a map of semisimplicial sets $Sing(X). \rightarrow Sing(Y).$, but I'm confused about how to do so. I guess the main confusion is ...
5
votes
1answer
94 views

Does the category of Simplicial Complexes have finite limits and colimits?

Does the category of Simplicial Complexes have finite limits and colimits? Does the geometric realization functor preserve them? Thanks!
3
votes
0answers
56 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 ...
4
votes
1answer
112 views

The two-sided simplicial bar construction is Reedy-cofibrant

Let $\mathcal{M}$ be a simplicial model category and let $\mathbb{C}$ be a small category. For a diagram $F : \mathbb{C} \to \mathcal{M}$ and a weight $G : \mathbb{C}^\mathrm{op} \to \mathbf{sSet}$, ...
3
votes
1answer
138 views

What is the cone over a simplicial set?

At the moment I'm reading through Edward B. Curtis, 'Simplicial Homotopy Theory' (Advances in Mathematics 6, 107-209 (1971)) in order to learn about simplicial sets and I run into a problem where the ...
0
votes
0answers
30 views

simplicial complex bijection [duplicate]

Given two compact Hausdorff spaces $X$ and $Y$ and $h \colon X \to Y$ a homeomorphism, how can I prove that $h_{\mathfrak{A}} : N(\mathfrak{A}) \to N(h(\mathfrak{A}))$ is a bijection where ...
4
votes
0answers
115 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
votes
2answers
187 views

'homotopy' between morphisms of a 'topological' or 'algebraic' category (Stanley-Reisner ring)

In what follows, a homotopy is a congruence $\simeq$ on a given category. Given such a homotopy, objects $X$ and $Y$ of the given category are homotopy equivalent when there exist morphisms ...
4
votes
2answers
131 views

Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ preserve (homotopy)colimits?

Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ given by $S(X)_m=Hom_{\mathbf{Top}}(\Delta^m,X)$ preserve colimits? If not, what is a counterexample? The only things I can say are that it ...
2
votes
1answer
123 views

What (filtered) (homotopy) (co) limits does $\pi_0:\mathbf{sSets}\to\mathbf{Sets}$ preserve?

Consider the functor $\pi_0:\mathbf{sSets}\to\mathbf{Sets}$. $\pi_0$ does not preserve arbitrary limits $\pi_0$ does not send homotopy limits to limits $\pi_0$ does preserve filtered colimits ...
1
vote
2answers
78 views

horn of a simplex

I'm reading one book of simplicial homotopy. It's just amazing. But I am stuck at the very beginning of the book. He let a simplicial set be a contravariant functor between the category $ \Delta $ of ...
6
votes
1answer
242 views

Toric Varieties: gluing of affine varieties (blow-up example)

Let $\Delta$ be a fan, consisting of cones $\sigma_0=conv(e_1,e_1+e_2)$ and $\sigma_2=conv(e_1+e_2,e_2)$ and $\tau=\sigma_0\cap\sigma_1=conv(e_1+e_2)$. The dual cones are $\sigma_0^\vee= ...
3
votes
0answers
64 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 ...
1
vote
0answers
48 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 ...
8
votes
2answers
333 views

What is the homotopy colimit of the Cech nerve as a bi-simplical set?

Let $f:Y_\bullet\to X_\bullet$ be an epimorphism of simplicial sets and define the bi-simplicial set $$ F_{\bullet\bullet}=\ldots Y\times_X Y\times_X Y\underset{\to}{\underset{\to}{\to}}Y\times_X Y ...
2
votes
2answers
93 views

What is a good way to simplicize the integer lattice?

I have a function $f$ defined on the positive integer lattice in $\mathbb{R}^n$ (i.e. the vectors in $\mathbb{R}^n$ whose coordinates are all nonnegative integers). I want to extend the domain of $f$ ...
0
votes
1answer
36 views

Paths between 0-cells in a classifying space. II

Let $\mathcal{C}$ be a small category and $X,Y$ objects within. If $X$ and $Y$ (as $0$-cells) lie within the same path-component in $B\mathcal{C}$, can one say anything in general on how $X$ and ...
1
vote
1answer
41 views

Paths between 0-cells in a classifying space.

Let $\mathcal{C}$ be a small category. Giving a morphism $u: X \to Y$ in $\mathcal{C}$ is equivalent to giving a functor $f: \{0 < 1 \} \to \mathcal{C}$ (with $f(0) = X$, $f(1) = Y$, and $f(0 \to ...
5
votes
1answer
149 views

What exactly is the CW complex structure on a geometric realisation?

This is likely a silly question. Definitions: $\bullet$ $\Delta_n = \{ (t_0, \dots, t_n) \: | \: 0 \leq t_i \leq 1, \sum_i t_i = 1 \}$ $\bullet$ Given $f: \underline{m} \to \underline{n}$ in ...
3
votes
0answers
47 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 ...
0
votes
0answers
44 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 ...
5
votes
0answers
140 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 ...
2
votes
1answer
81 views

Inner product between certain vectors on a simplex.

For $n\geq 2$, let $\Delta^n$ be a regular $n$-dimensional simplex in $\mathbb{R}^n$ centered at the origin ${\bf 0}$ and inscribed in the unit sphere $S^{n-1}$. Let ${\bf v}_0,{\bf v}_1,\ldots,{\bf ...
5
votes
1answer
115 views

Pseudomanifolds

An $n$-dimensional (closed) pseudomanifold is a finite simplicial complex $X$ such that (i) every simplex is a face of an $n$-simplex (ii) every $(n-1)$-simplex is a face of exactly two ...
3
votes
1answer
60 views

Random mixing of the space of triangulations of a surface

Summary: How quickly does the edge-flip random walk in the space of triangulations of a closed, connected, orientable surface converge to the uniform distribution over all triangulations? Let $M$ be ...
2
votes
1answer
140 views

adjunction relation

Assume I have simplicial sets $X:\Delta^{op}\rightarrow Set$ and $Y:\Delta^{op}\rightarrow Set$, then I can form the simplicial set $\operatorname{Hom}(X,Y)_n := ...
1
vote
0answers
72 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 ...
1
vote
1answer
82 views

Does the suspension functor preserve fibrations?

Let $X_{\bullet}$ be a simplicial set and let $\Sigma X_{\bullet}$ denote its simplicial suspension. If $X_{\bullet} \to Y_{\bullet}$ is a fibration, then is $\Sigma X_{\bullet} \to \Sigma ...
4
votes
0answers
200 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 ...
3
votes
0answers
78 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 ...
4
votes
2answers
146 views

The empty set in homotopy theoretic terms (as a simplicial set/top. space)

I am currently confused about the empty set in terms of its path components and how this fits into the Quillen adjunction between topological spaces and simplicial sets. Probably, one of my ...
8
votes
1answer
1k views

Difference between simplicial and singular homology?

I am having some difficulties understanding the difference between simplicial and singular homology. I am aware of the fact that they are isomorphic, i.e. the homology groups are in fact the same (and ...
1
vote
1answer
390 views

Understanding the definition and notation of geometric realization

I had trouble making sense of the definition of geometric realization of a simplicial set. Let $\Delta^n$ be the standard n-simplex defined as the functor $\hom_\Delta(-,$*n*$): \Delta \rightarrow$ ...
0
votes
1answer
75 views

How to show that $\Delta[n]$ isn't Kan fibrant…?

This is the problem: I have to prove that $\Delta[n]$ isn't Kan fibrant for n >=2. Does anyone how idea how to do it?
4
votes
0answers
110 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 ...
0
votes
0answers
53 views

Teminology for a “supercomplex”

If $\Delta_{1}$ and $\Delta_{2}$ are abstract simplicial complexes and $\Delta_{1}\subseteq \Delta_{2}$, then we say that $\Delta_{1}$ is a subcomplex of $\Delta_{2}$. Is there a terminology for ...
2
votes
1answer
49 views

Colimits of cosimplicial rings

The forgetful functor from the category of rings to the category of sets does not preserve arbitrary colimits. But it does preserve filtered colimits. For example, the colimit of a sequence of rings ...