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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Analogue of spectral sequences for simplicial sets

Is there an analogue to spectral sequences where, instead of chain (bi)complexes, we use simplicial sets? Namely, let $\{\mathbf{S}_{p,q}\}_{p,q}$ be sets such that $\mathbf{S}_{p,\bullet}$ and $\...
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30 views

How to define a transported version of the simplex $X = \{x \in \mathbb{R}^n_{+}| \sum\limits_{i = 1}^n x_i = 1\}$?

Let the simplex in $\mathbb{R}^n$ be denoted as $$X = \{x \in \mathbb{R}^n_{+}| \sum\limits_{i = 1}^n x_i = 1\}$$ So it looks like: I want to take a point $\bar x \in X$ (red dot) And drag it to ...
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14 views

Double Nerve Preserving Filtered Colimits

The double nerve (https://ncatlab.org/nlab/show/double+nerve) is a functor $\mathcal{N}_2:2Cat\rightarrow sSSet$ from the 3-category of 2-categories to the category of bisimplicial sets. Does the this ...
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34 views

geometric realization of a simplicial complex

Let $K$ be a simplicial complex and let $|K|$ be the geometric realization of $K$. Suppose that the vertices ($0$-simplices) of $K$ are $v_0,v_1,\cdots,v_k$, $k$ finite. Let $\Delta^n$ be the standard ...
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singular homology of a simplicial complex [duplicate]

On Page 108 of the book Algebraic Topology, Allen Hatcher, the singular homology of a topological space $X$ is defined to be the homology of the chain complex by setting the $n$-chains $C_n(X)$ as the ...
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14 views

Sheaves of Simplicial Rings?

Could someone provide me a reference for a treatment of sheaves of simplicial commutative rings? As in simplicial sheaves with a ring structure.
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1answer
48 views

A modern approach to homotopy theory in $\mathbf{SSet}$

I'm currently trying to understand the basics of homotopy theory for simplicial sets. However, my current sources (Peter Mays "Simplicial objects in algebraic topology" and Kans original "on c.s.s. ...
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12 views

A doubt regarding the definition of the boundary operator.

For any singular simplex $\sigma:\Delta_p\to X$, define a $(p-1)$- chain $\partial\sigma$ called the boundary of $\sigma$ by $$\partial\sigma=\sum\limits_{i=0}^p\sigma\circ F_{i,p}$$ The domain of $...
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1answer
52 views

CW complex structure of geometric realization

In Ralph Cohen's notes on the topology of fiber bundles he makes the following claims: on pp.69, he says the geometric realization of a simplicial set is a CW complex on pp.70, he says the geometric ...
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38 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 ...
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1answer
16 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 ...
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1answer
46 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 $$\hom_{\mathbf{S}}(X\times Y,Z)\simeq \hom_{\mathbf{S}}(Y,\mathbf{Hom}(X,Z))$$...
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1answer
42 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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14 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 ...
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30 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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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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16 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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28 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.
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65 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 ...
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1answer
92 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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16 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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41 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 ...
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1answer
70 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 ...
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1answer
29 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 ...
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61 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 ...
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1answer
54 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 ...
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15 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 ...
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1answer
27 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 $f_n:X_{n,\...
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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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1answer
56 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 $x=(x_0,...
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28 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 ...
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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
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 $k,j+1\neq i$. ...
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1answer
93 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
53 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 ...
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2answers
79 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 ...
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1answer
23 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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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
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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
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}...
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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 ...
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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 \...
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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,...
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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 ...
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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 ...
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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 ...
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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
56 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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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 ...
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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 $...