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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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 ...
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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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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 ...
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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 ...
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1answer
18 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 $k,j+1\neq i$. ...
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1answer
87 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
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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 ...
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2answers
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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 ...
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1answer
16 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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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
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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 ...
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1answer
63 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 ...
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1answer
31 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 ...
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1answer
34 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 ...
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31 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 ...
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1answer
48 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 ...
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1answer
21 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 ...
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2answers
96 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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25 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 ...
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1answer
39 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 ...
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33 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 ...
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45 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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1answer
37 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 ...
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1answer
67 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 ...
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1answer
36 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 ...
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37 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
71 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 ...
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1answer
73 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 ...
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40 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 ...
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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 ...
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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 ...
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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 ...
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25 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 $B$ ...
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1answer
36 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 ...
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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 ...
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1answer
64 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 ...
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1answer
53 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 ...
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11 views

n-regular hypergraph and its resolution

Let $G =(V(G),E(G)))$ be a $n$-regular hypergraph. Let $Z_n$ be a cylic group with generator $a.$ Definition : A resolution of $G$ is finite partially ordered set $C$ with $C_0$ is the set of ...
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22 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 ...
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$\Delta$-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 ...
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1answer
68 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. ...
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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 ...
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1answer
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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 ...
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68 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}) := ...
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1answer
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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 ...
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1answer
45 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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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. ...
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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 ...
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1answer
83 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 ...
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1answer
90 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 ...