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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3
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0answers
36 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 ...
6
votes
0answers
68 views
+50

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. ...
0
votes
1answer
19 views

Contradictory Orientations of Faces in Simplicial Complexes

From what I've read, an orientation of a simplex is an equivalence class of total orders on its vertices, where equivalence means up to even permutation. An orientation on an $n$-simplex induces an ...
2
votes
1answer
38 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 ...
5
votes
2answers
111 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 ...
6
votes
0answers
132 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 ...
0
votes
1answer
14 views

Relating maximal elements of downsets to minimal elements of the complement

Denote by $\mathcal{P}(S)$ the set of non-empty subsets of a finite $S$. Suppose that $A\subset \mathcal{P}(S)$ is a downset, i.e., every subset $Q$ of any $P\in A$ is also contained in $A$. We can ...
8
votes
3answers
452 views

What is combinatorial homotopy theory?

Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below. Many times I've heard people ...
5
votes
0answers
40 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 $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
1
vote
1answer
29 views

Examples of the Geometric Realization of a Semi-Simplicial Complex

I am reading The Geometric Realization of a Semi-Simplicial Complex by J. Milnor and here are the definitions: I find it difficult to visualize without specific examples. Can anyone help to provide ...
1
vote
1answer
45 views

homeomorphism between a boundary and a sphere

I started another question related to this. Consider $|\Delta^n|:=\{(x_0,..,x_n)\in\mathbb{R}^{n+1}:\sum_{i=0}^n x_i=1, x_i\in[0,1]\;\forall i\}$ the geometric realization of the standard n-simplex ...
2
votes
0answers
29 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
1answer
48 views

the 0-th homology of a simplical complex

Let $K$ be an (abstract) simplicial complex. The claim is: $H_0(K;\mathbb{Z})$ is always nonzero. Is this possible to prove it without any "special techniques to computing homology-groups"? ...
2
votes
1answer
31 views

$\sum_{i=0}^n (-1)^i \delta_i(\sum_{j=0}^{n+1} (-1)^j \delta_j) = 0$, given that $\delta_i \delta_j = \delta_{j-1}\delta_i$ whenever $i < j$

$\sum_{i=0}^n (-1)^i \delta_i(\sum_{j=0}^{n+1} (-1)^j \delta_j) = 0$, given that $\delta_i \delta_j = \delta_{j-1}\delta_i$ whenever $i < j$ This problem shows up in the middle of dealing with ...
0
votes
1answer
52 views
0
votes
1answer
60 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
votes
0answers
52 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
votes
0answers
31 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 ...
0
votes
1answer
40 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?
2
votes
0answers
25 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 ...
1
vote
1answer
15 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 ...
1
vote
1answer
409 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*$): \Delta \rightarrow$ ...
1
vote
1answer
36 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 ...
2
votes
0answers
60 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 ...
3
votes
0answers
50 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
votes
1answer
118 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 $\mathrm{Hom}_{\mathfrak{C}\Delta^n}(i, j) \simeq (\Delta^1)^{(j-i-1)}$, ...
3
votes
2answers
112 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 ...
0
votes
2answers
47 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 ...
0
votes
1answer
43 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
votes
1answer
122 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
votes
1answer
40 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 ...
6
votes
3answers
146 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 ...
1
vote
1answer
32 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 ...
0
votes
1answer
35 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
votes
1answer
55 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
votes
1answer
112 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 ...
0
votes
1answer
52 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.
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 ...
2
votes
1answer
399 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
votes
0answers
50 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 ...
1
vote
1answer
40 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 ...
1
vote
0answers
41 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 ...
2
votes
0answers
43 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
votes
1answer
759 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
votes
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
votes
0answers
90 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
votes
1answer
110 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
votes
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 ...
1
vote
0answers
45 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
votes
0answers
20 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 ...