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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1answer
42 views

Notation for a functor between comma categories

Suppose we have two categories $D$ and $S$, as well as two functors $K,L:D\to S$ and a natural transformation $\varphi:K\to L$. Given another category $C$ and a functor $Y:C\to S^D$, is there a nice ...
2
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0answers
37 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
26 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 ...
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1answer
31 views

cells of quotient CW complex

Let $X$ be a CW complex and $Y$ a CW subcomplex. If $X$ has no cell of dimension $n$, for some $n>0$, then $X/Y$ has no cell of dimension $n$. Is it true? Why?
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0answers
22 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 ...
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1answer
12 views

Skeletality of a simplicial set $X$ vs. highest degree of a non-degenerate simplex

Let $X$ be a simplicial set with a non-degenetate simplex in degree $n$ and suppose that all simplices in higher degrees are degenerate. Is $X$ an $n$-skeletal simplicial set and not ...
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1answer
379 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*$): \Delta \rightarrow$ ...
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1answer
32 views

Elementry collapses implies same homotopy type.

Let $\Delta$ be a simplicial complex, and suppose that $\sigma \in \Delta$ is a proper face of exactly one maximal simplex $\tau \in \Delta$. A simplicial collapse of $\Delta$ is the removal of all ...
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0answers
54 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 ...
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43 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 ...
2
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1answer
108 views

Realization (in the sense of homotopy coherent nerve) of $\partial\Delta^n$

I need some help understanding the working of the homotopy coherent nerve (as described in Lurie's HTT). Let $i < j$, then $\mathrm{Hom}_{\mathfrak{C}\Delta^n}(i, j) \simeq (\Delta^1)^{(j-i-1)}$, ...
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2answers
94 views

How do we get a simplicial homology functor?

The $n$-th simplicial homology group $H_n(A)$ of an abstract simplicial complex $A$ depends on the choice of an orientation for $A$ (but for different orientations, the homology groups are ...
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2answers
38 views

Null homotopic by simplicial approximation

If $m<n$ use the simplicial approximation theorem to prove that any map $f:S^m\to S^n$ is null homotopic. Deduce that $\pi_1(S^n)$ is trivial if $n>1$. we have not covered lot on simplicial ...
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1answer
41 views

Empty set in a simplicial complex

Should the empty set be considered a simplex in a simplicial complex? Which justifications exist for the answer? I guess it is somewhat comparable to $1$ not being a prime number.
4
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1answer
111 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 $G : \mathbb{C}^\mathrm{op} \to \mathbf{sSet}$, ...
0
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1answer
38 views

Simplicial complex of a graph?

Starting with a graph $G$, form a simplicial complex $X$ which has $G$ as the 1-skeleton, and then has higher dimensional simplices whenever more than two vertices of $G$ are mutually adjacent. So any ...
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3answers
124 views

Why is the 'mapping space' between two objects in a quasi-category a Kan complex?

Let $X$ be a quasi-category. (i.e. a simplicial set satisfying the weak Kan extension condition). Given two 'objects' $a,b\in X_0$ in $X$, one defines the mapping space $X(a,b)$ to be the pullback of ...
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1answer
29 views

Show that a set defines a simplicial complex

Let $A_n := \left\{-n,\cdots,-1,1,\cdots,n\right\}$ $\Delta_n := \left\{ B \subseteq A \; \big\vert \; \#(\{-i,i\}\cap B)\leq 1 \; \forall 1 \leq i \leq n \right\}$ Show that ...
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1answer
34 views

A question about abstract simplicial complexes and discs.

I find the following definitions in a book about algebraic topology: Definition: Let $K$ be an abstract simplicial complex. $(1)$ If $K$ is finite, simply connected and with nonempty ...
3
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1answer
47 views

Relative Simplicial Approximation

While I was studying the cellular approximation theorem on May's "A Concise Course in Algebraic Topology" I found something a bit unclear. I agree with the fact that, given two CW-complexes $X,Y$, and ...
8
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1answer
97 views

How to construct a quasi-category from a category with weak equivalences?

Let $(\mathcal{C},W)$ be a pair with $\mathcal{C}$ a category and $W$ a wide (containing all objects) subcategory. Such a pair represents an $(\infty,1)$-category. One model for such gadgets is a ...
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1answer
51 views

How does one give topological structure to an abstract simplicial complex?

Given an abstract simplicial complex $K,$ I'd like to know how I can endow it with a topology.
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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 ...
2
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1answer
368 views

Show simplicial complex is Hausdorff

I have a simplicial complex $K$ and I need to show that its topological realisation $|K|$ is Hausdorff. And $K$ need not be finite. I have very little idea on how to get started on this. Only that if ...
2
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0answers
48 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 ...
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1answer
39 views

Basic localizers contain adjoint functors

I'm struggling with a property of basic localizers, namely that adjoint functors are weak equivalences. Recall that a basic localizer $\mathcal W$ is a subset/subclass of the arrows of the category ...
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0answers
35 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 ...
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0answers
29 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 ...
4
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1answer
652 views

Barycentric subdivisions of simplices yield a simplicial complex

The following interesting result (in particular parts (b) and (d)) is stated either as a obvious fact or as an exercise in several books on algebraic topology: The barycenter $b_\sigma$ of an ...
0
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1answer
21 views

An elementary question about $\pi_0$ and two homotopic maps

Let $f\colon A\to B$ be a map of pointed simplicial sets and let $B$ be Kan-fibrant. Let $0\colon A\to B$ denote the map which factorizes over the basepoint of $B$. Is it true that $f$ is homotopic ...
7
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0answers
82 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 ...
3
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1answer
78 views

Why do we ask that condition in Kan complex?

Let $\{X_n\}_{n=0}^\infty$ be simplicial set with faces $d_i:X_n\to X_{n-1} $, a simplicial set is called a Kan Complex if for any $x_0,...,x_{k-1},x_{k+1},...,x_{n+1}$ if $d_i(x_j)=d_{j-1}(x_i)$ for ...
0
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1answer
27 views

The line between the centers of two k-simplices crosses through the centers of lower dimensional simplices

"The line between the centers of two k-simplices crosses through the centers of lower dimensional simplices." Is this always true? The barycentric centers are equidistant from the vertices. k=2 So ...
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0answers
38 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 ...
2
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0answers
19 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 ...
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0answers
18 views

Can one drop the retracts in the definition of anodyne extensions?

Definition: Anodyne extensions(i.e. acyclic cofibration) of simplicial sets are the closure of horn inclusions under transfinite composition, pushouts, and retracts. The composition and pushouts can ...
4
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1answer
52 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 ...
6
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1answer
96 views

Cohomological Whitehead theorem

Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...
5
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1answer
145 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 ...
2
votes
1answer
117 views

Product of simplicial complexes?

Given two abstract simplicial complexes $\mathcal{K}$ and $\mathcal{L}$, what is the definition of their product $\mathcal{K} \times \mathcal{L}$, as another abstract simplicial complex? Basically I'm ...
3
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1answer
59 views

Does $\pi_0$ preserve infinite products?

The functor $\pi_0\colon \operatorname{sSets}\to \operatorname{Sets}$ has value on a simplicial set $X$ the coequalizer of the two boundary morphisms $d_0,d_1\colon X_1\to X_0$. It is easy to see ...
5
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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!
1
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1answer
56 views

Simplicial n-sphere as a coequalizer? (trivial question)

I know that it is a trivial question. And it can be found in any text on simplicial sets. (So, sorry about that). But I'm studying the Jardine's Book on Simplicial Homotopy Theory (Jardine and ...
3
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1answer
73 views

n-truncated simplicial set

It might be a trivial question. So, I apologise in advance. Let $ \Delta ^{op}_n $ be the full subcategory of $ \Delta ^{op} $ such that the set of objects of $ \Delta ^{op}_n $ is $ \left\{ 0, ...
4
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1answer
91 views

Existence and homotopies of embeddings between simplicial complexes

Let $K$ and $L$ be simplicial complexes, $m=\dim K$, and $h:|K|\rightarrow |L|$ be a continuous map. Show that $h$ is homotopic to a map carrying $K$ into $L^{(m)}$, the $m$-skeleton of $L$. I'm ...
3
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2answers
59 views

Why is the Cech nerve $C(U)$ of a surjective map $U\to X$ weakly equivalent to $X$?

Let $f:U\to X$ be a surjective map of sets and $$ ...U\times_XU\times_XU \substack{\textstyle\rightarrow\\[-0.6ex] \textstyle\rightarrow \\[-0.6ex] ...
3
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1answer
52 views

Difference between the simplicial nerve and the nerve of a simplicial category

In Jacob Lurie's Higher Topos Theory book, he defines the following notion of a simplicial nerve: Definition 1.1.5.5. Let $\mathcal{C}$ be a simplicial category. The simplicial nerve ...
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2answers
62 views

What is the equivalent of Delaunay tringulations in high dimensions?

For 2D manifolds, Delaunay triangulation is a very useful tool for coarse graining. It has the nice property that in the flat/euclidian manifold case, it reduces to a 2D simplicial tesselation of the ...
4
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1answer
85 views

Model structure on sSet

Which is the model structure on $ \text{sSet} $ (category of simplicial sets) that makes $\text{sSet}$ Quillen equivalent to the category $ \text{Cat} $ (of small categories) by the adjunction ...
2
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0answers
32 views

What is a homotopy between bisimplicial maps

I am looking for the naive notion of homotopy. For maps $f, g: A\to B$ of simplicial sets $\Delta^{op}\to Sets$, a homotopy $H$ is a simplicial map $H:A\times\Delta^{1}\to B$ such that ...