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.
773 questions
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An efficient description of simplicial sets as colimits of standard simplices?
In general, any presheaf $X$ is a colimit of representables indexed by $\int X$. In this proposition from Kerodon, Lurie refines this as follows: if the simplicial set $S$ has dimension $\leq k$ then ...
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Simplicial Approximation
I have simplicial complex $\tau$ which is defined as
$\tau := sd^2\mathbb{S}$ where $\mathbb{S} \subset [0,1]$ such that
$$
\mathbb{S} = \bigl\{ \langle0\rangle, \langle1\rangle, \langle0,1\rangle \...
2
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1
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Trivial cofibrations of simplicial sets, which are bijective on vertices
Let $f\colon K\rightarrow K^{\prime}$ be a map of simplicial sets that is a trivial cofibration (i.e. a monomorphism and a weak homotopy equivalence), which is bijective on vertices. Is $f$ in the ...
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cocartesian Fibrations
I need to show that that a cocartesian fibration between $\infty$-categories $p: C\to D$ s.t all fibers are equivalent to $\Delta^0$ is an acyclic Kan fibration.
I am really not sure how to do that. I ...
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Does any surjective morphism from an affine hypercovering of the pro-etale site to a w-contractible one admit a section?
I am reading the paper "The Pro-Etale Topology for Schemes" by Bhatt-Scholze and am trying to understand Remark 4.2.9. Unless otherwise said, all results I reference are from this paper. ...
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1
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subinfinity categories
Hey I have the following question. Let $C,D$ be two infinity categories (i.e simplicial sets with inner horn filling condition). $D$ is a subsimplicial set of $C$ if we just take some simplices from $...
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Reference for function space definition of geometric realization of abstract simplicial complex
The following is described on Wikipedia as one way to define the geometric realization of an abstract simplicial complex $K$:
The construction goes as follows. First, define $|K|$ as a subset of $[0,...
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Can I use simplicial homology mod 2 for topological data analysis?
I’ve started working on my masters project in TDA (topological data analysis) and I’m currently getting to grips with simplicial homology. I can’t seem to get a straight answer online, so I thought I’...
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These boundary homomorphisms are represented as matrices: How do I compute the image and homology groups induced by them?
I'm implementing my own program for topological data analysis (TDA). I'm pretty familiar with the basic theory of algebraic topology, so my issue is mainly regarding the implementation side of things.
...
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composition on simplicially enriched categories.
Let $\mathcal{C}$ be a simplicially enriched category i.e a Functor: $\Delta^{op}\to Cat$ with constant objects. Let $x,y \in ob(\mathcal{C}[n])$. We want to define a simplicial set $Hom_{\mathcal{C}}(...
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Is there any particular reason why I need a multiset for persistence barcodes? Can I define a barcode this way instead?
I've been reading the introductory paper written by these UPenn reseachers on topological data analysis, and in particular I've been focused on their section of persistent homology. To round things ...
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1
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Triangulation of N datapoints in $\mathbb{R}^d$
Given a set of $N$ data points in $\mathbb{R}^d$, is a triangulation always composed of the same number of "triangles", i.e. $d$-simplices?
This seems to be the case when $d=2$. The number ...
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1
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Kan complexes and $S$ functor
In our lecture we constructed the functor $S: Top \to sSet$ which sends $X \mapsto \hom_{Top}(\Delta^{\bullet},X)$. We proved that $S(X)$ is always a Kan complex, however I was wondering if also every ...
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Simplicial homotopy
I need to show that for a Kan complex $Y$ simplicial homotopy of maps $X \to Y$ rel $A$ is a equivalence relation.
I found a proof for that in nLab https://ncatlab.org/nlab/show/simplicial+homotopy ...
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Poset whose Localization is a Finite Cyclic Group
According to the answers to this question, every group can be realized as the localization of a partially ordered set at some subcategory of weak equivalences. However, I've been unable to construct ...
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Homotopy equivalence of realization of nerve of a cover.
I was reading the paper "Homology fibration and group completion theorem" by Segal and McDuff. During the proof of Proposition 6 (of local-to-global flavor) he states the existence of spaces ...
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Trying to understand $\widetilde{W_\alpha}$ during the universe construction in simplicial sets?
I am reading https://arxiv.org/abs/1211.2851 'The Simplicial Model of Univalent Foundations (after Voevodsky)' and I have problems to understand construction of $\widetilde{W_\alpha}$ that is being ...
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The model categories of $\mathrm{sSet}$ and $\mathrm{CGWH}$ are Quillen equivalent
By $\mathrm{CGWH}$, we mean the category of compactly generated weakly Hausdorff topological spaces, and by $\mathrm{sSet}$, we mean the category of simplicial sets. I am seeking a reference for the ...
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Moore complex of opposite simplicial set
I have often seen the claim that different conventions for the normalized Moore complex interact in a simple way with taking opposites of the simplicial abelian group at issue. But I haven't been ...
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Euler characteristic and the Mertens function?
I am interested if there can be given any applications of this topology on prime powers and a sheaf on it, to number theoretic questions ( I am looking for known results in elementary number theory, ...
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1
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What are cylinder objects in the category of simplicial sets?
I think the projection $X\times\Delta^1\to X$ is a cylinder object in sSet the category of simplicial sets (with the Quillen model structure), since $|X\times \Delta^1|\cong |X|\times_k [0,1]\to |X|$ ...
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1
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Décalage as a model for a path space object?
I am wondering how one can view a (plain) décalage of a simplicial space (or at least a simplicial set) $X$ as its path space object in the sense of the model category theory. Although this is what I ...
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Prove continuity of the affine extension mapping between geometric simplicial complexes
Let $\Delta_1$ and $\Delta_2$ be geometric simplicial complexes. Let $K_1$ and $K_2$ be their associated abstract simplicial complexes. Let $f: V(K_1) \to V(K_2)$ be a simplicial mapping.
We define ...
- 1
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1
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Cosimplicial resolution associated to a monad
Let $\mathcal{C}$ be a category and $\mathbf{T}$ a monad on $\mathcal{C}$ with functor part $T : \mathcal{C} \to \mathcal{C}$ (I would actually like to consider the case where $\mathcal{C}$ is an $(\...
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On the induced morphism between colimits in an $\infty$-category
Let $\mathcal{C}$ be an $\infty$-category (quasicategory), $F\colon K\rightarrow\mathcal{C}$ a diagram in $\mathcal{C}$ and $y$ a colimit of $F$. To be precise, there is a colimit cone $\overline{F}\...
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Reference for a proof on homotopy groups of simplicial abelian groups
In Simplicial Homotopy Theory, Goerss and Jardine wrote :
The simplicial abelian group structure on A induces an abelian group structure
on the set $\pi_n$(A, 0) = [($\Delta^n$, $\partial \Delta^n$), ...
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45
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Free abelian group functor preserves Kan fibration
Suppose, $\mathbb{Z}$ is the free abelian group functor from simplicial sets to simplicial abelian groups. Then does $\mathbb{Z}$ preserves Kan fibrations i.e. if $X \to Y$ is a Kan fibration between ...
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2
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Naturality for the Homotopy Fiber Sequence of Mapping Spaces
For a cartesian fibration $p\colon\mathcal{E}\rightarrow\mathcal{C}$ of $\infty$-categories (quasicategories) and objects $x,y$ of $\mathcal{E}$, the induced map $\mathrm{map}_{\mathcal{E}}(x,y)\...
- 14.6k
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simplicial commutative rings and derived commutative rings
I met two definitions with regard to the simplicial(or derived) commutative rings. One way is very direct and literal, that is, a 'simplicial ring' is a simplicial object in the category of ring：
\...
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Nerve theorem and cycles deformation
Consider a finite set of point $P\subset[0,1]^{d}$. Let $r>0$, by nerve theorem, we know that, the Čech complex $C^{2r}(P)$ and $B_{2}(P,r)$ are homotopy equivalent.
Furthermore, Proposition 3.2 of ...
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Infinite category structure on SCRing and 'space of commutative squares' in SCRing
When I read this paper 'virtual cartier divisors and blow ups', I often meet with such phrase like 'mapping space of infinite category $SCRing_{A}$'. See lemma 2.3.5 in the above paper: $Map_{SCRing_{...
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What morphism is sent to a monomorphism by the left Kan extension ${\rm Lan}_{\Delta}\colon{\bf Set^\Delta\to\bf Set^{\hat\Delta}}$ along Yoneda?
For any small category $C$, let us write $\hat{C} = \mathbf{Set}_C$ for the presheaf category $\mathbf{Set}^{C^{\mathrm{op}}}$, and $y=y_C\colon C\to \mathbf{Set}_C$ for the Yoneda embedding. Consider ...
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The core of an $\infty$-category and pointwise invertible maps $\Delta^1 \times \partial \Delta^n \cup \{1\} \times \Delta^n \to \underline \hom(B,X)$
I'm currently working through Theorem 3.5.11 of Cisinski's Higher Categories and Homotopical Algebra. The section in which the theorem is inscribed concers the core of an $\infty$-category.
Trying to ...
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1
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How to understand May's proof that counit map is a weak equivalence?
A similar question was asked about 4 years ago here, but received no answers, so I hope it is appropriate to post a new question. I am trying to read the singular homology section in May's Concise ...
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1
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Inclusion of a certain full subcategory of the category of elements of a simplicial set is a final functor
For a simplicial set $X$, let $\text{el X}$ be its category of elements, whose objects are pairs $([n], x\in X_n)$. Let $\text{(el X)}_{nd}$ denote the full subcategory comprising objects $([m], y\in ...
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Simplicial $\pi_0$ as homotopy classes $\Delta^0 \to X$ using that $\pi_0$ is left-Quillen
In Higher Categories and Homotopical Algebra, Cisinksi defines the connected component functor as
$$\pi_0 \colon \mathsf{sSet} \to \mathsf{Set}, \qquad \pi_0(X) = \mathsf{colim}_{n} X_n,$$
the left ...
- 17.3k
2
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1
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Does geometric realization commute with finite limits?
I am trying to find out if geometric realizations i.e. the functor $|-|\colon\text{sSet}\to \text{Top}$ commutes with finite limits. In the following post the user claim that this is well known:
https:...
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Definition for map $T$ is unclear? - Hatcher pg. 122
This comes from Hatcher's Algebraic Topology textbook (pg. 122). See photo below.
(Note: $b_\lambda([w_0,...,w_n])=[b_\lambda,w_0,...,w_n]$ where $b_\lambda$ defines the barycenter of singular $n$-...
- 1,054
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Geometric intuition behind induced homology homomorphisms
This is certainly overkill...but at the moment, I'm trying to strengthen my geometric understanding towards homology groups. The current trouble I've been having is in understanding the geometric ...
- 1,054
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1
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Linearity assumption on $f_{\#}$? (Hatcher, pg. 110-111)
This is coming from pg. 110-111 of Hatcher's Algebraic Topology textbook. (See below photos.) At the moment, it is unclear why we are able to simply construct $f_{\#}$ such that linearity is ...
- 1,054
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1
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Reference request for realizing a simplicial set as the homotopy colimit of its simplices
I know that
$$X\simeq hocolim_{Simp(X)}\Delta^n,$$
where $Simp(X)$ is the category of simplices of $X$, I know this for example because of proposition 7.5 of the nLab's page for homotopy limits. ...
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1
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The geometric realization of a simplicial set does not determine it
I read in Gallauer's notes on infinity categories, in "Warning 1.9." without proof, that the geometric realization of a simplicial set does not determine it and wanted to make sure I ...
- 1,280
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3
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Showing that the inclusion of the 2-skeleton of a simplicial complex induces an isomorphism on fundamental groups
Suppose that $K$ is a simplicial complex and let $K_{(2)}$ denote its 2-skeleton.
Show that that for all $x \in |K_{(2)}|$ the map
$i_*: \pi_1(|K_{(2)}|,x) \to \pi_2(|K|,x)$ induces an isomorphism of ...
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0
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57
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Cofibrations and cofibrant objects in a simplicial abelian category
For context, I am reading chapter III.2 of Goerss and Jardine's book on simplicial homotopy theory, but I am not an expert in model categories. In that chapter, they introduce the model structure on ...
- 11
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1
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Morphism inducing isomorphism in homotopy category is weak equivalence
In Goerss-Jardine's book "Simplicial Homotopy Theory" they prove in Lemma 4.1 of chapter II that for a simplicial model category $\mathcal{C}$ the statement
$$\text{ If }f:X\to Y \text{ in } ...
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Describe all cones in $\mathbb R^n$ that are simplicial and cosimplicial.
For any finite set of vectors $(v_1, \ldots, v_k \in \mathbb{R}^n)$, let's define:
$Cone(v_1,…,v_k)= \{λ_1v_1+…+λ_kv_k,∣,λ_1,…,λ_k \ge 0\}$ and $Poly(v_1,…,v_k)= \{ x \in \mathbb{R}^n ∣(v_i, x) \ge 0, ...
- 1
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1
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71
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Weak equivalence of filtered Colimit
Given a model category $C$, I have two functors $F,G:\mathbb{N}\rightarrow C$, where see $\mathbb{N}$ as sequence category.
Question: Given a natural transformation $J:F\rightarrow G$ and suppose the $...
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0
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40
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Realization of singular sets preserving homeomorphism type
Let $X$ be a topological space. The simplicial set $\operatorname{sing}(X)$ has as its $n$ simplicies all singular $n$-simplices $\Delta^n \to X$. The realization of the singular simplicial set is ...
- 2,063
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1
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Segal subdivision
I call Segal subdivision the endofunctor of simplicial objects in a category $\mathcal{C}$ induced by the doubling endofunctor of $\Delta^{op}$ sending $x_0<\cdots<x_n$ to $x_0<\cdots <x_n&...
- 1,579
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Triangulating Product of Simplicial Complexes
I am currently working on a problem for which I believe the following result is crucial.
The result of this problem was discussed in this post.
Product of simplicial complexes?
However it is not ...
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