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# 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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### Identifying the delaunay simplex the point resides in without computing all the possible simplices

My question is whether it is possible to identify the delaunay simplex the point $\mathbf{x}_{\textrm{query}}$ resides in without pre-calculating all the possible simplices by triangulation? Can we ...
64 views

### Eilenberg-Zilber's lemma existence

Let $X$ a simplicial set, $x\in X_m$ an $m$-simplex. We say that $x$ is degenerate if $\exists s:[m]\to[n]$ an epimorphism such that $n<m$ and $y\in X_n$ such that $X(s)(y)=x$. Now I want to show ...
1 vote
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### Epimorphism in simplex category is split

Consider $\Delta$ the simplex category, with objects $[n]=\{0,\dots,n\}$ and morphisms $f:[n]\to [m]$ such that $i<j\implies f(i)\leq f(j)$ (my definition is with $i<j$). I have shown that ...
61 views

### Let K be a simplicial complex and σ a simplex of K. Show that the link lk(σ) is a subcomplex of K, if lk(σ) is not empty.

I wanna know if I choose the right way to do my proof? it's correct? To show that the link $\textrm{Lk}(\sigma)$ is a subcomplex of $\mathcal{K}$, if $\textrm{Lk}(\sigma)$ is not empty, we need to ...
1 vote
46 views

### $\mathfrak{h}(\mathcal{G}^{\Delta^1})$ is the category of commutative squares in $\mathfrak{h}\mathcal{G}$

This is related to the question Homotopy category of a symmetric monoidal $(\infty,1)$-category is symmetric monoidal. Let us fix an $(\infty,0)$-category $\mathcal{G}$ (a Kan complex). We want to ...
95 views

### Is every $\Delta$-complex realizable as Simplicial complex, ie triagulazable?

In the following I will use definitions and constructions of simplicial complexes and $\Delta$-complexes/sets from Greg Friedman's An elementary illustrated Introduction to simplicial sets. I'm ...
60 views

### Understanding the Beck-Chevalley condition (II)

The older question here on the site asks about the intuitive meaning of the Beck-Chevalley condition. Accidentally one of the answers has caught my eye. It made me wonder if it is possible to describe ...
53 views

### "Compatible /admissible" maps of $\Delta$-complexes

In the following I'm going to use definitions and constructions of simplicial complexes and $\Delta$-complexes/sets from Greg Friedman's An elementary illustrated Introduction to simplicial sets. ...
283 views

### Where does one learn how to apply categorical algebra and higher abstractions to algebraic topology?

Tl;Dr: I know higher category theory and algebra is used ubiquitously in advanced algebraic topology. However, every time I ask someone, or try to find out, how one actually learns to apply the higher ...
104 views

### An infinity-category is a functor, but is it a category in some way and is the simplex notions so universal?

For some time I had the impression that infinity-categories are just a generalization of higher order categories: categories whose arrows have arrows among them, and they have further arrows etc. ...
71 views

### Geometric realization of a simplicial set is Hausdorff

I am reading the book May's Simplicial objects in algebraic topology and I am trying to understand the proof for Theorem 14.1 - the geometric realization of a simplicial set is a CW-complex. He says ...
1 vote
57 views

### Proving that the geometric realisation of a minimal fibration is a Serre fibration - have I got the details right?

$\newcommand{\O}{\mathcal{O}}$In the book: "Simplicial Homotopy Theory", by Goerss-Jardine, they 'prove' that every minimal fibration $q:X\to Y$ of simplicial sets $X,Y$ has $|q|:|X|\to|Y|$ ...
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1 vote
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### "Small" simplicial group equivalent to the circle

Let $G$ be a topological group, call $\mathcal{H}\in sGrp$ simplicial model of $G$ if there is a homomorphism of topological groups $|\mathcal{H}|\to G$ which is also a homotopy equivalence. In a ...
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### Difficulty using the (co)limit formulae to construct the $n$-(co)skeleton left and right Kan extensions for truncated simplicial objects

Tl;Dr - I’m struggling to show that the $n$-skeleton is a Kan extension, from the basic limit formula (this should be possible, as it was “left to the reader” in my book). I’m also struggling to even ...
1 vote
64 views

### What is the takeaway of cofibrant generation?

I have recently begun reading about model categories. In particular, I have been using Balchin's A Handbook of Model Categories as a reference, and the following quote has been quite perplexing. A ...
1 vote
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### Is the inclusion of the "smooth singular Kan complex" a homotopy equivalence?

Let $X$ be a smooth manifold. Let $L$ be the singular simplicial set of $X$. We know $L$ is a Kan complex. Let $K \subset L$ be the "simplicial subset" consisting of only those singular ...
1 vote
56 views

### Elementary properties of the "forgetful" functor from simplicial sets to semisimplicial sets?

Would anyone know a reference which explains whether the forgetful/restriction functor from simplicial to semisimplicial sets is, or fails to be: full, faithful, essentially surjective, etc?
80 views

### Is there a "cubical" version of quasicategories?

Definitions: A Kan complex is a simplicial set having the horn-filling condition; A quasicategory is a simplicial set having the inner-horn-filling condition. My question is: I was wondering if ...
67 views

### Would anyone know an example of a simplicial set having the "spine-lifting" property while not being a quasicategory?

Definition: By a simplicial set we mean a Set-valued presheaf on the category $\mathbf{\Delta}$ of nonempty finite totally-ordered-sets with non-strict order-preserving functions between them. The ...
1 vote
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### Could anyone give an example of a quasicategory which does not have the "spine lifting" property?

Definition: By a simplicial set we mean a Set-valued presheaf on the category $\mathbf{\Delta}$ of nonempty finite totally-ordered-sets with non-strict order-preserving functions between them. The ...
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### Calculating Colimits in The category of Simplicial Sets

I am struggling to understand how to take colimits in the category of simplicial sets. To make my notation clear, I will use $\delta_i : [n-1]\to[n]$ to denote the $i$-th face map (it picks out the ...
1 vote
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### Universal Property of The Simplex Category Δop

I tried formulating the universal property of the simplex category. There seemed to be two different options to choose for the universal property of $\Delta^{op}$. I'm having trouble making ...
### Calculating the limit over $\partial \Delta ^2$
Suppose we have the diagram of spectra $$A \to B \to C$$ Then taking the limit of this diagram is essentially taking the limit over the Nerve of the category $$\cdot \to \cdot \to \cdot$$ which I ...
I'm studying "Introduction to $\infty$-categories", by Markus Land, and I found myself in need of constructing simplicial maps from their values on non singular simplices. This is what I'd ...