# 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
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### 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 ...
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### 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 ...
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### 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-...
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### 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 ...
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### 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!
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### 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. $CF^n=F([n])$...
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### 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 ...
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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 $... 0answers 151 views ### Unique representation of a degenerate simplex I recently learned about simplicial sets, and now I'm studying some basic properties. I've learned that every degenerate simplex is a degeneracy of a unique non-degenerate (nice) simplex. However, I ... 2answers 145 views ### Are bounded open regions in$\mathbb{R}^n$determined by their boundary? Let$U$and$V$be two bounded open regions in$\mathbb{R}^n$, and let us further assume that their topological boundaries are nice enough that they are homeomorphic to finite simplicial complexes. ... 1answer 646 views ### Why do we ask that condition in Kan complex? Let$\{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)$for$...
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I'm having trouble following the proof of Proposition I.5.2 in Goerss-Jardine (Simplicial Homotopy Theory). After establishing the adjunction $\hat\Delta(X\times K,Y) \simeq \hat\Delta(K,[X,Y])$, ...
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### 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 ...
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### 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 ...
### 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?
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: ...