Questions tagged [simplicial-stuff]

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-Kan correspondence etc. Please do not use for questions about geometry of simplices nor about triangulations.

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46
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1answer
8k views

Difference between simplicial and singular homology?

I am having some difficulties understanding the difference between simplicial and singular homology. I am aware of the fact that they are isomorphic, i.e. the homology groups are in fact the same (and ...
16
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2answers
456 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 ...
13
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3answers
1k 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 speaking ...
13
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0answers
536 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 $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
10
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3answers
802 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 ...
10
votes
1answer
3k views

Product of simplicial complexes?

Given two abstract simplicial complexes $\mathcal{K}$ and $\mathcal{L}$, what is the definition of their product $\mathcal{K} \times \mathcal{L}$, as another abstract simplicial complex? Basically I'm ...
10
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1answer
175 views

Higher categories for category theorists?

I'm not actually a category theorist, but assume I have some background in category theory and am pretty comfortable with it; and I want to learn higher category theory (say in the sense of Boardmann-...
10
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0answers
156 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 ...
9
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3answers
519 views

Right Kan extension of $\mathcal{F} : \mathsf{\Delta} \rightarrow \mathsf{Top}$.

My question arose while studying something about Kan Extensions. We know that we have the following diagram $$ \begin{array}{ccc} &&\mathsf{\Delta} & \xrightarrow{\mathcal{F}} & \...
9
votes
1answer
742 views

Fat geometric realization weakly equivalent to the usual one

Let $X$ be a simplicial set. Recall that we can associate to it a certain topological space, the geometric realization, given by $ | X | = \int ^{n \in \Delta} X_{n} \times \Delta _{n} \simeq (\...
9
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1answer
786 views

Toric Varieties: gluing of affine varieties (blow-up example)

Let $\Delta$ be a fan, consisting of cones $\sigma_0=conv(e_1,e_1+e_2)$ and $\sigma_2=conv(e_1+e_2,e_2)$ and $\tau=\sigma_0\cap\sigma_1=conv(e_1+e_2)$. The dual cones are $\sigma_0^\vee= conv(e_2,e_1\!...
8
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2answers
863 views

What is the homotopy colimit of the Cech nerve as a bi-simplical set?

Let $f:Y_\bullet\to X_\bullet$ be an epimorphism of simplicial sets and define the bi-simplicial set $$ F_{\bullet\bullet}=\ldots Y\times_X Y\times_X Y\underset{\to}{\underset{\to}{\to}}Y\times_X Y \...
8
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1answer
651 views

Does the category of Simplicial Complexes have finite limits and colimits?

Does the category of Simplicial Complexes have finite limits and colimits? Does the geometric realization functor preserve them? Thanks!
8
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1answer
373 views

How does hocolim relate to Hom?

In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like this:$\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\colim}{colim}\DeclareMathOperator{\hocolim}{hocolim}\...
8
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1answer
187 views

A Mayer-Vietoris argument I don't get — From a paper by Bousfield-Gugenheim

I working through the article On PL De Rham theory and rational homotopy type by Bousfield-Gugenheim. I am having troubles completing the proof of lemma 5.3, where they claim that the fact that a ...
8
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1answer
220 views

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. ...
8
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1answer
98 views

When are the geometric realisations of two simplicial sets homeomorphic?

I'm given two simplicial sets $X,Y : \Delta^{\operatorname{op}} \to Set$. Of, course, if I study their geometric realisations $\lvert X \rvert$ and $\lvert Y \rvert$, I might find a homeomorphism, or ...
8
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1answer
154 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 ...
8
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0answers
279 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 ...
7
votes
3answers
1k views

Kan fibrations and surjectivity

I have a basic question on the usual model structure on simplicial sets. What is the relation between being a Kan (trivial maybe ?) fibration and surjectivity ? Surjectivity here means either ...
7
votes
1answer
301 views

Boundary of a simplicial set in terms of a coequalizer

I am trying to understand why we have a coequalizer $\sqcup_{0 \leq i < j \leq n} |\Delta^{n-2}| \rightrightarrows \sqcup_{0 \leq i \leq n} |\Delta^{n-1}| \rightarrow |\partial \Delta^n|$. What ...
7
votes
1answer
378 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}...
7
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1answer
536 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 ...
7
votes
1answer
466 views

Homology other than singular?

Usually, one defines $n$-th homology functor on topological spaces as the composite functor $$ \mathbf{Top} \to [\Delta^\mathrm{op},\mathbf{Set}] \to [\Delta^\mathrm{op},R\!-\!\mathbf{Mod}] \overset C ...
7
votes
1answer
77 views

Homotopy (co)limit of sets

There are general constructions for homotopy (co)limit of diagrams in spaces $$D:I \rightarrow Top$$ I was wondering if the homotopy (co)limit of discrete diagrams, i.e. when the spaces are all ...
7
votes
2answers
421 views

Cohomological Whitehead theorem

Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...
7
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0answers
60 views

Is the nerve of a symmetric monoidal category a K-theory space?

It's an amazing fact that if the $C$ is a symmetric monoidal category so that its components form a group, then $NC$ is an infinite loop space. Now if we have a Waldhausen category $D$, a category ...
7
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0answers
126 views

Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the simplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a point?...
7
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0answers
242 views

What is a copresheaf on a “precategory”?

Let $\mathcal{S}$ be a category with pullbacks. A precategory $\mathbb{C}$ in the sense of Borceux and Janelidze [Galois Theories, §7.2] comprises three objects $C_0, C_1, C_2$ in $\mathcal{S}$ and ...
6
votes
1answer
771 views

What does it mean for a category to be “tensored over” another category?

What does it mean for a category to be "tensored over" another category? I was reading "Stable model categories are categories of modules" by Schwede and Shipley (http://www.math.uni-bonn.de/people/...
6
votes
1answer
884 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 ...
6
votes
2answers
291 views

How to intrinsically think about simplicial objects.

It seems that a simplicial set should be thought as a space/thing, with encoded building information. The set of $n$-simplicies is the set of n-dimensional pieces and the boundary and degeneracy maps ...
6
votes
1answer
836 views

Coboundary Formula

When $K$ is a simplicial complex, the dual complex $C^*(K)$ to the chain complex $C_*(K)$ has a concrete interpretation: an element in $C^n(K)$ (a cochain) is given by assigning an integer to every ...
6
votes
2answers
335 views

Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ preserve (homotopy)colimits?

Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ given by $S(X)_m=Hom_{\mathbf{Top}}(\Delta^m,X)$ preserve colimits? If not, what is a counterexample? The only things I can say are that it ...
6
votes
1answer
73 views

Issue with Coequalizer Definition of the Horn of a Simplex

I am having trouble understanding the maps in this coequalizer defining the k-horn (from page 9 of Goerss and Jardine's Simplicial Homotopy Theory). It is defining them using the ith, jth inclusion ...
6
votes
1answer
138 views

$\infty$-categories definition disambiguation

I am currently reading Lurie's - Higher Topos Theory, but am having difficulties from the very beginning. I know that the topic is too broad and different mathematicians may define differently some of ...
6
votes
1answer
201 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
1answer
485 views

What are simplicial topological spaces intuitively?

(NOTE: I reposted the question to MO. Please answer there.) I tend to imagine simplicial objects in a category as some kind of "topological objects", with a notion of homotopy. Simplicial sets are ...
6
votes
1answer
496 views

What exactly is the CW complex structure on a geometric realisation?

This is likely a silly question. Definitions: $\bullet$ $\Delta_n = \{ (t_0, \dots, t_n) \: | \: 0 \leq t_i \leq 1, \sum_i t_i = 1 \}$ $\bullet$ Given $f: \underline{m} \to \underline{n}$ in $\...
6
votes
1answer
151 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. $CF^n=F([n])$...
6
votes
1answer
1k views

What is the cone over a simplicial set?

At the moment I'm reading through Edward B. Curtis, 'Simplicial Homotopy Theory' (Advances in Mathematics 6, 107-209 (1971)) in order to learn about simplicial sets and I run into a problem where the ...
6
votes
1answer
252 views

Inverse functor in proof of Dold Kan Correspondence

I´m looking at the proof of the Dold-Kan correspondence. Let $SA$ be the category of simplicial objects in an abelian category $A$ and $CH_{\ge0}(A) $ the category of non-negative chain complexes in $...
6
votes
0answers
151 views

Unique representation of a degenerate simplex

I recently learned about simplicial sets, and now I'm studying some basic properties. I've learned that every degenerate simplex is a degeneracy of a unique non-degenerate (nice) simplex. However, I ...
5
votes
2answers
145 views

Are bounded open regions in $\mathbb{R}^n$ determined by their boundary?

Let $U$ and $V$ be two bounded open regions in $\mathbb{R}^n$, and let us further assume that their topological boundaries are nice enough that they are homeomorphic to finite simplicial complexes. ...
5
votes
1answer
646 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 $...
5
votes
2answers
555 views

Passing pullbacks through adjunction

I'm having trouble following the proof of Proposition I.5.2 in Goerss-Jardine (Simplicial Homotopy Theory). After establishing the adjunction $\hat\Delta(X\times K,Y) \simeq \hat\Delta(K,[X,Y])$, ...
5
votes
2answers
658 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 ...
5
votes
1answer
507 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 ...
5
votes
1answer
292 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?
5
votes
2answers
284 views

Equivalent definition for a collection of simplices to be a simplicial complex

I am reading the following lemma from Munkres' Elements of Algebraic Topology: Lemma 2.1 A collection $K$ of simplices is a simplicial complex if and only if the follow hold: ...

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