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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83 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
1answer
67 views

Realization of simplicial sets

Consider a simplicial set $X$, i.e. a contravariant functor from $\mathbf{\Delta} \to \mathbf{Set}$ where $\mathbf{\Delta}$ has as objects $[n]:=\{0, \cdots, n \}$ for all $n \in \mathbb{N}$ and as ...
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
67 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 \mathbf{sSet}...
1
vote
1answer
45 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 \...
1
vote
1answer
51 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 ...
4
votes
2answers
360 views

What's stopping me from choosing the nth Eilenberg Mac Lane space to be the following simplicial abelian group?

Given an abelian group $X$, let $F_n(X)$ denote the simplicial abelian group defined as follows: $F_n(X)_j=0$ for all $j<n$ and $F_n(X)_j=X$ for all $j\geq n$ with the appropriate zero and ...
2
votes
2answers
100 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 ...
0
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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 approximation,...
1
vote
1answer
53 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 ...
4
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1answer
231 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 ...
1
vote
1answer
24 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 ...
0
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0answers
51 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 $...
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0answers
34 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 ...
0
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0answers
37 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 ...
3
votes
1answer
53 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
80 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 $\...
0
votes
1answer
59 views

Why $|N(P_{i, j})| \cong [0, 1]^n$ as stated in page 21 of HTT?

maybe this is an idiot question, however I could not solve this after thinking for a while. I added the tag about higher categories simply because of the nature of the question, however this is just a ...
2
votes
1answer
41 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
40 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 ...
2
votes
1answer
82 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 ...
1
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1answer
126 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
votes
0answers
49 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
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0answers
65 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
65 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 ...
0
votes
0answers
14 views

How to visualize barycentric subdivision of higher order simplices?

I understand the idea of barycentric subdivision for 1-simplex and 2-simplex. However for higher simplices like 3-simplex or 4-simplex, how do we visualize the barycentric subdivision for those ...
0
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0answers
28 views

Relative Barycentric Subdivision question

I have been stuck at this question for days, partly because I don't know how to visualize Barycentric Subdivision of high dimensional simplexes. Question: Let $L$ be a simplicial complex and let $B$ ...
3
votes
0answers
53 views

Classifying space of resolution of a n-regular hypergraph

Let $G =(V(G),E(G)))$ be a $n$-regular hypergraph. Let $Z_n$ be a cyclic group with generator $a.$ Definition : A resolution of $G$ is finite partially ordered set $C$ with $C_0$ is the set of ...
1
vote
1answer
47 views

Fiber of fibration of simplicial sets

$\require{AMScd}$ If $p:E\to B$ is a fibration of simplicial sets, is the fiber in the model category sense, i.e. the homotopy limit of $$\begin{CD}{} @. E \\@. @VVV \\*@>>> B \end{CD}$$ the ...
1
vote
1answer
68 views

Topology on the simplicial complex consisting of the edges $\{1,n\}$ for $\ n \in \mathbb{Z}$

Let $X$ be the simplicial complex consisting of the edges $\{1,n\}$ for $\ n \in \mathbb{Z}$, and $Y=\{re^{in}: 0\leq r\leq 1, n\in \mathbb{Z}\}$. These 2 spaces look similar but I guess they are not ...
1
vote
1answer
638 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
0answers
25 views

Lifting a homotopy class $S^k\to X$ into a simplicial set $X$ which is not fibrant but satisfies some weaker horn filling condition

Let $X$ be a connected simplicial set. If $X$ is an Kan complex and $k\geq 0$, then every element $$ \tilde f\in\operatorname{Hom}_{Ho(sSet)}(S^k,X) $$ of the homotopy classes from $S^k$ to $X$ lifts ...
5
votes
1answer
69 views

Cech Cohomology and the Dold-Kan Correspondence

Given a (co/contravariant) functor $F$ from the simplicial category $\Delta$ to an abelian category $A$, we can form its Cech complex (or "alternating face map complex" on the nLab), i.e. $CF^n=F([n])$...
2
votes
0answers
49 views

$\Delta$-Complexes Are Hausdorff

I am using the definition of a $\Delta$-complex as given in Hatcher's book here on pg 103. Now on pg. 104, just before the section on Simplicial Homology, Hatcher remarks that if $X$ has a $\Delta$-...
2
votes
0answers
83 views

How to compute (co)limits of enriched categories?

Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$. This ...
1
vote
0answers
83 views

On two definitions of the nerve of a simplicial category

Let ${\mathcal C}$ be a simplicial category. Then there are the following two ways of constructing a simplicial set from ${\mathcal C}$: Form the simplicial nerve $\text{N}_\Delta({\mathcal C}) := \...
2
votes
1answer
38 views

explicit equivalent relation in the expression of the classifying space of a monoid

Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. (It's also called the internal nerve.) The ...
1
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1answer
57 views

what means 'the realization of a topological category'

In the paper Homology Fibrations and the "Group-Completion" Theorem. page 280 bottom line 10-bottom line 12, what means 'the realization of a topological category'?
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1answer
28 views

homotopy module of a simplicial module

I'm reading a paper about the cotangent complex and I'm having trouble with one of the definitions (3.4 of http://homepages.math.uic.edu/~bshipley/iyengar.pdf ). Let $V$ be a simplicial $R$-module. ...
6
votes
1answer
123 views

cofaces and codegeneracies on Simplicial Sets

Let Δ be the category of finite ordinal numbers with order-preserving maps, i.e., Δ consists of objects strings A morphism f:n→[m] is an order-preserving function (a functor) and we can think of the ...
2
votes
1answer
89 views

Induced map on simplices from order-preserving maps between finite ordinal numbers

Let $\Delta$ be the category of finite ordinal numbers with order-preserving maps, i.e., $\Delta$ consists of objects strings $[n]: 0→1→2→⋯→n$. A morphism $f:[n]→[m]$ is an order-preserving function (...
3
votes
1answer
71 views

Universality of the Simplex Category. Proving Functoriality of the Map.

Let $B$ be a strict monoidal category, and $\left \langle c,\mu ',\eta ' \right \rangle$ a monoid in $B$. Now suppose we consider the simplex category $\left \langle \triangle ,+,0 \right \rangle$, ...
0
votes
1answer
109 views

Category of ordinal numbers

Let $\Delta$ be the category of finite ordinal numbers with order-preserving maps, i.e., $\Delta$ consists of objects strings $$ [n]: 0 \to 1 \to 2 \to \dots \to n. $$ A morphism $f:[n] \to [m]$ is ...
7
votes
1answer
103 views

Boundary of boundary of singular cube is zero (Spivak)

At the bottom of page 99 of M. Spivak's Calculus on Manifolds he arrives at the formula $$\partial (\partial c)=\sum_{i=1}^n \sum_{\alpha=0,1} \sum_{j=1}^{n-1} \sum_{\beta=0,1} (-1)^{i+\alpha+j+\beta}...
2
votes
0answers
40 views

Representation of functions in the simplex category

I am using the following characterization of the simplex category $\Delta$: The objects are finite ordinals and the arrows are weakly monotone functions $f:n\rightarrow m$. $0$ is initial and $1$ is ...
0
votes
1answer
49 views

Is the skeleton-coskeleton adjunction $sSet$-enriched?

Let $n\geq 0$ be an integer. Is the adjunction $$ \mathbf{sk}_k\colon sSet \leftrightarrows sSet\colon\mathbf{cosk}_k $$ of the skeleton and coskeleton an $sSet$-enriched adjunction?
0
votes
0answers
55 views

Classifying space infinite totally ordered set contractible

Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This should be the (possible infinite) simplex with vertices $X$, hence I ...
3
votes
1answer
92 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?
3
votes
1answer
134 views

Homology of a simplicial set

Let $X$ be a simplicial set. Define the complex $(C^X_\bullet,D)$ by $$C^X_n=\bigoplus_{X_n} \mathbb{Z}$$ and $$D_n=\sum_{i=0}^n (-1)^i d_i:C_n \to C_{n-1}$$ where the $d_i$'s are the face maps. I ...
2
votes
1answer
63 views

Why is the internal hom of a Kan complex also a Kan complex?

Let $X, Y : \Delta^{op} \to \text{Set}$ be simplicial sets. I've seen it stated that if $Y$ is a Kan complex, then the internal hom $Y^X$ is a Kan complex also. How does one show this?