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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5
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0answers
31 views

Replacing faces of a cube in a quasicategory

I have a question on quasicategories which seems to be unavoidably cubical, and which I haven't been able to locate any information about. Suppose I have a cube $U:\mathbf{2}^3:\to C$ in a ...
1
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1answer
15 views

Enriching categories of simplicial objects

Let $C$ be a cocomplete category and $Simp(C)=C^{\triangle^{op}}$ the category of simplicial objects in $C$. I want to show that $Simp(C)$ is simplicially enriched but I don't understand how the ...
2
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1answer
36 views

Simplicial homotopy's exponential law

The following was cited from Simplicial Homotopy Theory by John F. Jardine and Paul Gregory Goerss. We have an adjunction $$\hom_{\mathbf{S}}(X\times Y,Z)\simeq ...
4
votes
1answer
38 views

Question about limit of cosimplicial diagram associated with a sheaf

Let $F$ be a presheaf of sets on the usual topological site. Let $\left\{U_i\rightarrow U\right\}$ be covering of $U$. On the second page of Hypercovers and Simplicial Presheaves the authors write the ...
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0answers
12 views

Ratio of dimension sizes in a pure $2$-dimensional simplicial complex

I'm listening to Alex Lubotzky's YouTube lecture on Cohomology and Computer Science, and during a proof he makes the claim: $\displaystyle\frac{|X(2)|}{|X(1)|} \sim \frac{n}{3}$, where $n$ is the ...
4
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0answers
26 views

The dimension of $H_i(X_n, \mathbb{Q})$ is linear in $n$ with a bounded nonnegative error, where the error is periodic?

Let $X$ be a finite simplicial complex with a simplicial map to $S^1$. Take covers of $X$ associated to the subgroups $n \mathbb{Z}$ of $\mathbb{Z} = \pi_1(S^1)$. This defines a sequence of spaces ...
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0answers
7 views

pointed simplicial set as coequalizer

Im studying simplcial sets and homotopy theory. I found this statement that seems quite immediate but for me it is not. Let $X$ be a pointed simplicial set, then $X$ can be realized as the ...
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0answers
11 views

Question on subgroup cohomology restricting proper, simplicial actions of an algebraic group

I have a question regarding an assertion made in p. 2 of these notes on Bruhat-Tits buildings. The question concerns the group $G_p=SL_n(\mathbb{Q}_p)$ and its subgroup $\mathbb{Z}^{n-1}$ (the ...
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0answers
25 views

Intuitive meaning for Kan fibration

Other than the formal definition of Kan fibration (https://en.wikipedia.org/wiki/Kan_fibration), is there any intuitive meaning of Kan fibration? Thanks.
3
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0answers
63 views

Can some Lie groups ($S^3$ in particular) be converted to simplicial groups?

I'm trying to understand the concept of Kan simplicial sets and simplicial groups in particular. My question is, in a nutshell: What is the relation between the group operation in a simplicial ...
3
votes
1answer
79 views

classifying space of a category

In his K-book, chapter IV, Weibel states the following as a “a straight forward application of Van Kampen's Theorem”: Lemma 3.4 Suppose that $T$ is a maximal tree in a small connected category ...
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0answers
11 views

Computation of the Moore Complex

Given a simplicial abelian group G, one can compute its simplicial homotopy groups as the homology groups of its associated Moore complex. Can someone please show me the explicit computation of the ...
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0answers
22 views

Is this a triangulation of a cylinder?

I am currently a beginner in Algebraic topology. I don't know whether triangulations of a thing are unique or not. So I thought to ask here whether the "triangulation" I've come up with is really a ...
4
votes
1answer
64 views

What is a “cubical map” between cubical complexes?

What is a natural definition of a cubical map between cubical complexes? What is its geometric realization? I found some definitions, such as here or here, where a cubical map between cubical ...
1
vote
1answer
25 views

Understanding semisimple Z modules

I'm trying to understand the nature of semisimplicity in Z modules. So for instance would I be right in thinking $Z/30Z \oplus Z/2Z$ is semi simple as it can be expressed as $Z/3Z \oplus Z/5Z \oplus ...
4
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0answers
53 views

Tensoring a connective chain complex with a simplicial set

Let $\mathrm{Ch}_{\geq 0}(R)$ be the category of chain complexes of $R$-modules concentrated in nonnegative degrees, equipped with the projective model structure. By a general theorem about model ...
0
votes
1answer
48 views

Homotopy category of a simplicial category

In many places (for example here) I've seen the following definition: For a simplicial category $\mathcal{C}$, it's homotopy category is defined to be the category $Ho(\mathcal{C})$ with the same ...
0
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0answers
14 views

Faces of a Bipyramid over a a Simplicial Polytope

Is there a simple way of expressing the number of faces of a bipyramid built over a polytope that is known to be simplicial, using the number of faces of the original polytope? This seems an easy ...
0
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1answer
24 views

Converse of realisation lemma for bisimplicial sets

Given two bisimplicial sets $X_{\bullet,\bullet}$ and $Y_{\bullet,\bullet}$, we have the result that if given a map $f:X_{\bullet,\bullet}\to Y_{\bullet,\bullet}$ such that the restriction ...
2
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0answers
29 views

Cohomology of geometric realization of a simplicial topological space

Let $X$ be a simplicial topological space. We can consider to notions of cohomology of $X$. Denote by $|X|$ geometric realization of $X$. Then we can take just $H^*(|X| )$ (i.e. usual cohomology of ...
1
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1answer
49 views

Example of Spherical Element (Simplicial Homotopy)

Definition: An element $x\in X_n$ is said to be spherical if $d_i x=*$ for all $0\leq i\leq n$. $X$ is a pointed fibrant simplicial set. I am puzzling over this definition. For instance, if ...
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0answers
27 views

A Grothendieck topology on $\Delta$

Is there a choice for a Grothendieck topology on $\Delta$ for which most interesting simplicial sets are sheaves (like representables, horns and boundaries, and more generally all categories)? I ...
3
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0answers
46 views

Fibrant (Kan complex) geometric meaning

A simplicial set $X$ is said to be fibrant (or Kan complex) if it satisfies the following homotopy extension condition for each $i$: Let $x_0,\dots,x_{i-1},x_{i+1},\dots,x_n\in X_{n-1}$ be any ...
0
votes
1answer
19 views

Matching faces in Simplicial Set theory

Let $X$ be a simplicial set. The elements $$x_0,\dots,x_{i-1},x_{i+1},\dots,x_n\in X_{n-1}$$ are said to be matching faces with respect to $i$ if $$d_jx_k=d_kx_{j+1}$$ for $j\geq k$ and $k,j+1\neq i$. ...
3
votes
1answer
92 views

Degeneracies of simplex $y$ which appears as any face of some simplex $x$

Let $K$ be simplicial set and $d_i:K_n\rightarrow K_{n-1}$, $s_i: K_n\rightarrow K_{n+1}$ ($i = 0,...,n$) face and degeneracy maps respectively. Suppose we have some $x\in K_n$ with $d_0x = ... = ...
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2answers
46 views

Is there any expression to calculate the homology groups of a quotient space?

Let $B \subset A$ where $A$ is a topological space and $A/B$ the space obtained from $A$ via collapsing $B$ to a single point. I was wondering if there is any expression for $H_k(A/B)$ in terms of ...
2
votes
2answers
67 views

How can I calculate the homology group of an infinite torus using Mayer-Vietoris?

I want to calculate the (simplicial) homology of the following space using Mayer-Vietoris: I have tried to do it by cutting it along the axis and getting two subspaces homeomorphic to something ...
2
votes
1answer
21 views

Why does sheaf $\pi_0$ of a simplicial presheaf determine maps to sheaves?

This question refers to an argument from Section 6, p. 22 of Freed–Hopkins, "Chern–Weil forms and abstract homotopy theory." There's something presumably straightforward I'm just not ...
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0answers
7 views

Geometric Realisation of a cyclic map

could somebody please explain me why the geometric realisation of cyclic maps is $S^1$-equivariant? This is e.g. used in the proof of 7.3.11 in Loday's cyclic homology Thanks
0
votes
0answers
28 views

understanding a statement in Weibel's “The K-book” about bisimplicial sets

Here is a theorem from Weibel's The K-book, Chapter IV Theorem 3.6.1. Let f : X → Y be a map of bisimplicial sets. (i) If each simplicial map $X_{p,∗} → Y_{p,∗}$ is a homotopy equivalence, so is ...
2
votes
1answer
64 views

On the join of simplicial sets as a dependent product

Prop. 3.5 of Joyal notes in quasicategories gives a description of $X\star Y$ as $i_*(X,Y)$, where $i^*\dashv i_*$ is the adjunction $$ i^*\colon \mathbf{sSet}/\Delta[1] \leftrightarrows ...
1
vote
1answer
40 views

Example for geometric realization on semi-simplicial set that doesn't preserve limit

I'm looking for a diagram $D$ (as simple as possible) in the category of semi-simplicial sets (i.e $sSet$ with only monos) such that $R(\text{lim}\,D) \ncong \text{lim}\,R(D)$, where $R$ is the ...
1
vote
1answer
42 views

Geometric realisation of simplicial map

Typically texts give a good definition of the geometric realisation $|\Delta|$ of a simplicial complex $\Delta$. I'm supposing that the geometric realisation forms a functor $|-|:\text{sComp} \to ...
0
votes
0answers
34 views

Simplicial Approximation Concrete Diagram (Experts needed)

Let $K$ and $L$ be simplicial complexes as given, and let $\phi:|K|\to|L|$ be the continuous map, where $A=\phi(a)$, $B=\phi(b)$, and so on. Check whether the map $\phi$ has a simplicial ...
1
vote
1answer
50 views

definition of a $\kappa$-small simplicial set

Let $\kappa$ be a regular cardinal. A set $X$ is $\kappa$-small, if $|X|<\kappa$. What does it mean for a simplicial set $X\colon \Delta^{op}\to Sets$ to be $\kappa$-small. I can imagine at ...
1
vote
1answer
23 views

homology of a specific complex

Consider the complex $\lbrace 1, 2, 3, 4, 12, 23, 34, 41, 13, 123 \rbrace$. Visually, it's a square with a diagonal edge, with one hole and one face. In computing the homologies, I ended up getting ...
2
votes
2answers
99 views

Milnor's proof of $|X\times Y| \cong |X|\times |Y|$

I'm on my way reading this article of Milnor about the geometric realisation of a "(complete) semi-simplicial complex" ( = simplicial set, in modern terms). I encountered a problem in a passage of ...
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0answers
33 views

Geometric realization of a “simplicial space up to homotopy,” part two

This question is a follow-up, and my initial motivation for asking Is there a sensible way to form the geometric realization of a "simplicial space up to homotopy"? Given that the questions ...
2
votes
1answer
52 views

Is there a sensible way to form the geometric realization of a “simplicial space up to homotopy”?

I am new to simplicial methods, and have some naive questions. As such, the questions may be malformed and the terminology might not even be right. I assume these are answered in some reference and ...
0
votes
0answers
35 views

Help with formalisation of a reasoning about simplicial sets

We briefly mention during lectures that the homotopy relation between two maps of simplicial sets is not necessarily symmetric. So I thought for a counterexample but I don't know how to formalise it ...
0
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0answers
48 views

Is the unique map $\emptyset\to \Delta^0$ really a horn inclusion? [duplicate]

Is it true that $\emptyset \to \Delta^0$ should be considered a horn inclusion? This seems to imply some terrible things, unless I'm missing something. In particular, it implies that a Kan fibration ...
3
votes
1answer
42 views

Geometric realization of the product of two simplicial sets

Let $X$ and $Y$ be two simplicial sets. Then there is a natural continuous map $\lvert X \times Y \rvert \to \lvert X \rvert \times \lvert Y \rvert$, where the right-hand side is given the product ...
3
votes
1answer
74 views

Nerve of a groupoid is a cyclic set

On page 28 of the COCTALOS lecture notes, it is written that a simplicial set is the nerve of a groupoid if and only if it is a cyclic set in the sense of Connes (I am understanding a presheaf on ...
2
votes
1answer
39 views

Examples of finite simplicial sets

Let $K$ be a simplicial set. A simplex $x\in K_{n}$ is said to be non degenerate if it is not the degenerancy of a $n-1$ simplex, i.e if there is no $y\in K_{n-1}$ such that $s_{i}y=x$. A simplicial ...
0
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0answers
38 views

homotopy between constant simplicial sets

Assume that $K,L$ are constant simplicial sets (i.e all the faces maps and the degenerancies maps are equal to the identity and $K_{i}=K_{0}$, $L_{i}=L_{0}$ for $i>0$). Assume that there is a ...
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1answer
78 views

Initial strict monoidal category

nlab provides a universal property of the cube category $\Box$. Definition. The cube category is the initial strict monoidal category $(M,\otimes,I)$ equipped with an object $int$ together with ...
2
votes
1answer
93 views

Intuition for the Dold-Kan correspondence

maybe this question does not make sense and it's just a psychological problem of mine. However I cannot understand the geometric picture of the Dold-Kan correspondence. Let $\mathbf{Ab}$ denotes the ...
0
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0answers
45 views

Infinitely “deep” globular sets / categories

In the usual sense a globular set consists of objects, arrows between objects, 2-arrows between arrows, etc. having $n$-arrows for every $n \in \mathbb N$ (although some may be empty). A category (or ...
2
votes
0answers
59 views

Question on the relation between the $n$-coskeleton functor and the internal hom-adjunction

(First, let me apologize that I asked an unanswered and related question in the stable case.) The category $\operatorname{sSet_+}$ of pointed simplicial sets is symmetric monoidal closed, i.e. there ...
0
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0answers
63 views

Understanding commutative cochain problem

I have just read about commutative cochain problem(CCP) here and I'm trying understand it. It states that you cannot turn(in nontrivial way) simplicial set $S$ to differential graded commutative ...