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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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1answer
28 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 ...
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1answer
31 views

Tensor product of exact complexes is exact

Let $M_\circ = \dots \to M_n \dots \to M_0 \to 0$ and $N_\circ = \dots \to N_n \dots \to N_0 \to 0$ be exact complexes of modules over a ring $A$ such that each module is flat. Is it then true that $(...
1
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1answer
33 views

Simplicial Sets as a Lawvere Theory

A Lawvere theory is a category $T$ with finite coproducts in which every object is isomorphic to a finite coproduct $\amalg_{i = 1}^n x$ of a distinguished object $x$. Then a $T$-theory is a functor $...
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1answer
54 views

Pushout-product of anodyne extensions is again anodyne

Currently, I'm reading Simplicial Homotopy Theory by Jardine and Goerss and I got stuck in the proof of the theorem about pushout-products of anodyne extensions (corollary 4.6 in the book). Namely, I ...
3
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2answers
71 views

There's an equivalence of simplicial categories $\Delta \to \tilde{\Delta}^{\text{op}}$.

Let $\Delta$ be the simplicial category, that is, the category of finite totally ordered sets and order-preserving maps. Let $\tilde{\Delta}$ be the subcategory where objects are those of $\Delta$ ...
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1answer
22 views

How to see that $S(\sigma) = \text{Hom}_{\Delta}(\sigma, [0,1])$ maps to the category $\tilde{\Delta}^{\text{op}}$.

Let $\Delta$ be the simplicial category. Let $\tilde{\Delta}$ be the subcategory of non-empty totally ordered sets as objects and order-preserving maps that also preserve the smallest and largest ...
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1answer
56 views

Let $K$ be a simplicial complex (that need not be finite). Prove $|K|$ is Hausdorff.

As the title makes clear, I'm trying to solve a question which asks me to show the topological realisation of a simplicial complex is Hausdorff. The question reminds me that a subset of $|K|$ is open ...
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0answers
34 views

Classifying Space of a Category Contractible [duplicate]

My question refers to a statement in Laures' and Szymik's "Grundkurs Topologie" (page 233). Sorry, there exist only a German version. Here the relevant excerpt: My question is why a category having a ...
2
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0answers
41 views

If matching maps of a cosimplicial space are fibrations up to degree $n$, does the $n$-th partial totalization have the 'correct' homotopy type?

Let $X$ be a cosimplicial space and suppose that: the matching maps $$ s:X^{k}\rightarrow M^{k-1}X $$ are fibrations for all $k\leq n$. On one hand we can form the partial totalization $Tot_n(...
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1answer
39 views

The normalized chain complex of a simplicial set

The paper The Geometry of Rewriting Systems, by K. Brown, mentions the notion of a normalized chain complex associated to a simplicial set. How is this complex defined? (and perharps the non-...
2
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1answer
100 views

Horn of a Simplicial Set

My question refers to description of horns $\Lambda^n_k$ for Kan Fibrations in Laures' and Szymik's "Grundkurs Topologie" (page 227). Sorry, there exist only a German version. Here the relevant ...
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1answer
29 views

Realization of Simplicial Sets respects Product

My question refers to an argument in the the proof of thm 11.6 in Laures' and Szymik's "Grundkurs Topologie" (page 227). Sorry, but there exist only a German version. Here the relevant excerpt: The ...
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0answers
28 views

Lemma for Hurewicz Theorem (Bredon)

I am trying to understand the following lemma: If $f,g:I\rightarrow X$ are paths s.t. $f(1)=g(0)$ then the 1-chain $f*g-f-g$ is a boudary. Proof: On the standard 2-complex (should it say simplex?) $...
3
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1answer
49 views

Explicit model of the $E_{n}$-operad in simplicial sets

The space of $k$ little $n$-disks, denoted $E_{n}(k)$, is usually constructed in the category of topological spaces as the space of $k$-tuples $(c_{d_{1},p_{1}},\dots, c_{d_{k},p_{k}})$ of disjoint $n$...
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0answers
38 views

Mapping spaces for pro-objects in a simplicial model category

If $C$ is a simplicial model category (I have simplicial commutative rings in mind), then one can simplicially enrich pro-$C$ over sSet by taking $$\lim_j \mathrm{colim}_i \underline{\mathrm{Hom}}_C(...
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1answer
36 views

Simplicial set associated to a topological space

Let $X$ be a topological space. We want to associate a simplicial set i.e., a functor $\Delta\rightarrow \underline{\text{Set}}$. Here $\Delta$ is the category whose object set is$ \{[0],[1],[2],\...
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1answer
34 views

Does Singular complex determine a topological space

This question comes from thinking about the singular geometric realization adjunction $Sing \dashv |\cdot|$. I suspect that this adjunction is not monadic, so using the Monadicity theorem I tried to ...
2
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0answers
42 views

Degeneracies for semi-simplicial sets

Consider the category of semi-smplicial sets, ie simplicial sets without degeneracies. It is a preseaf category over $\Delta_{inj}$, the category of ordinals together with injective maps. The usual ...
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0answers
24 views

Factorisation into cofibration and trivial fibration

Suppose $M$ is a simplicial model category, $f: R \to T$ a morphism between fibrant objects in $M$. Is there a nice way to construct a factorization of $f$ into a cofibration followed by a trivial ...
2
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0answers
73 views

Is there a natural homotopy inverse to the map $∣Sing(X)∣\rightarrow X$

Let $X$ be an object of the category $C$ of CW complexes. Let $Sing:C \rightarrow SSet$ be the functor sending a CW complex to its singular complex (a simplicial set). Let $∣-∣: SSet \rightarrow C$ ...
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0answers
54 views

Is this a 'relative' colimit or some other categorical construction?

Let $\mathcal{A}$ be a closed covering of some space $X$, then let $\Sigma[\mathcal{A}]$ be the category of intersections of subsets of $\mathcal{A}$, further let $F: \Sigma[\mathcal{A}] \to \mathsf{...
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0answers
40 views

Need help understanding comment in Higher Topos Theory

I am having trouble understanding this part in Lurie's Higher Topos Theory. This can be found in section 2.4.4.4 right after Lemma 2.4.4.1. Lemma 2.4.4.1. Let$ p : \mathcal{C} \rightarrow \mathcal{...
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0answers
35 views

Can nonisomorphic groupoids have homotopy equivalent classifying spaces?

We know that two discrete groups having the same classifying space up to homotopy are isomorphic. One can just take fundamental groups and conclude. The situation with topological groups is subtler. ...
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1answer
42 views

Using simplicial homology to compute topological spaces homeomorphic to the quotient spaces of polygons

I am reading Introduction to Algebraic Topology by Rotman and the following is presented as a method to use simplicial homology to compute topological spaces that are homeomorphic to the quotient ...
3
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1answer
55 views

Does there exist a higher-dimensional 5-sided “tetrahedron + 1”?

The first shape is "0"-sided and is a point. The next shape is a line segment and it's "1"-sided. The next shape is a triangle and it's 3-sided. The next shape is a tetrahedron and it's 4-sided. ...
3
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1answer
63 views

Initial objects on $\infty$-categories

Let $X \in \mathbf{Set}_{\Delta}$ an $\infty$-category and $\tau_1$ the left adjoint functor to the nerve $\mathrm{N} \colon \mathbf{Cat} \to \mathbf{Set}_{\Delta}$. Show that if $x$ is an initial ...
2
votes
1answer
80 views

What is the fundamental category?

Given a category $\mathcal{C}$, we have a nerve functor $$\mathrm{N} \colon \mathbf{Cat} \to \mathbf{Set}_{\Delta}$$ that assigns to $\mathcal{C}$ its nerve $\mathrm{N}(\mathcal{C})$. This functor ...
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0answers
69 views

About a Kan fibration (Postnikov towers for simplicial sets)

I am trying to understand a specific construction of Postnikov towers for simplicial sets, as explained for instance here (under "absolute Postnikov tower") So you start with a simplicial set $X$ (I ...
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0answers
35 views

Join of simplicial sets induces a functor.

Given a simplicial set $X$, denote by $(\mathrm{Set}_{\Delta})/_X$ the over category, whose objects are morphisms of simplicial sets with source $X$. I want to show that the joint $X \star Y$ of ...
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1answer
71 views

Why is induced map on zero homology the identity and not negative the identity?

Suppose we have a simplicial map $f$ on a path connected simplicial complex $X$. The answer here: Induced map on zeroth homology is zero claims that the induced map on the $0$-homology given by $f_*: ...
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0answers
93 views

Group cohomology topologically with simplicial sets

I have a question about the usual formula for the differential in the usual projective resolution of $\mathbb{Z}$ as a $G$-module for a finite group $G$ Recall that for a $G$-module $A$, $C^i(G,A)$ ...
3
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1answer
63 views

Simplicial set as a colimit

Let $K \in \mathrm{Set}_{\Delta}$ be a simplicial set. Then $K$ is the colimit of the diagram $$F \colon \Delta/K \to \mathrm{Set}_{\Delta} \, $$ that assigns to each $\Delta^n \to K$ the ...
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0answers
46 views

Which simplicial sets are filtered colimits of standard simplices?

The question is all in the title : every simplicial set is a colimit of the standard simplices $\Delta^n$, but I'm wondering which ones are filtered or directed colimits of these, if there's a nice ...
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0answers
96 views

Theorem 2.1.2.2 Higher Topos Theory

At the page 74 of HTT, there is the following theorem Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....
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0answers
47 views

Higher homotopical information in racks and quandles

A quandle is defined to be a set $Q$ with two binary operations $\star,\bar\star\colon\ Q\times Q\to Q$ for which the following axioms hold. Q1. a $\star$ a = a Q2. (a $\star$ b) $\bar\star$ b = (a $...
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1answer
24 views

Notation for Geometric realization of simplicial sets

I am confused about some notation in the quick way of constructing the geometric realization of a simplicial set. Consider the simplex category $\Delta \downarrow X$ of a simplicial set $X$. The ...
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0answers
30 views

Axioms for the $\Delta$-complex and orientation problem

A $\Delta$-complex structure on a space $X$ is a collection of maps $\sigma_{\alpha}:\Delta^n \to X $,with n depending on the index $\alpha$,such that: 1.The restriction $\sigma_{\alpha}|\...
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1answer
21 views

Is $Cat_{\Delta}$ enriched over itself?

Question is just as in the title. Is the category of simplicially enriched categories enriched over itself? If not, is it enriched over another relevant category, e.g., simplicial sets?
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0answers
20 views

Weak complicial set from strict $2$-category

I am reading this paper: https://arxiv.org/pdf/1610.06801.pdf It says: If $C$ is a strict $2$-category, there is a unique saturated $2$- trivial complicial structure on $NC$, in which the $2$-cell ...
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0answers
33 views

$\mathbf{sSet}$-enriched Algebraic Theories

If $\mathcal{L}$ is the $\mathbf{sSet}$-enriched subcategory of $\mathbf{sSet}$ whose objects are finite coproducts of the terminal simplicial set $\Delta^0 = \Delta(-,[0]) = *$, identify the object $\...
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0answers
41 views

Yoneda lemma for standard simplicial n-simplex

Let $\Delta$ denote the "ordinal number category" whose objects are categories $\bf{n}$ where $\mathbf{n}=0\to1\to2\to\cdots\to n$ where identity morphisms are suppressed. A morphism in $\Delta$ is $\...
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2answers
98 views

What are classifying spaces of algebraic categories like?

Let $\mathcal{C}$ be a category. Recall that the nerve $N(\mathcal{C})$ of $\mathcal{C}$ is the simplicial set obtained by defining $N(\mathcal{C})_k$ to be the set of $k$-tuples of composable ...
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1answer
43 views

Finite connected poset is contractible

Given $P$ a finite connected poset is this contractible? In my intuition the answer would be yes: I cannot really picture such a poset which is not contractible. I know that if $P$ has a maximum or ...
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0answers
27 views

Reference of material regarding both simplicial stuff and HoTT

I am currently reading both the HoTT book and "Stuff on quasicategories" by Rezk, and recently I am feeling that some thoughts of them looks similar. Could someone please point out something to let me ...
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1answer
28 views

n-skeleton product preserving.

It is true that the n-skeleta functor $$sk_{n}:sSet\rightarrow sSet $$ is product preserving i.e., $sk_{n}(X\times Y)$ is naturally isomorphic to $sk_{n}(X)\times sk_{n}(Y)$.
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1answer
37 views

Two-fold branched cover of Hexagon - Which map is meant

I am trying to understand an obvious example in a paper but do not get what is meant by: "X is a hexagon and $f:X \rightarrow \sigma^2$ is a two-fold branched cover (branched at the center of $\sigma^...
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1answer
46 views

map on connected components is injective

Consider the usual model structure on $SSet$ (which is proper). Let $X$ and $Y$ be two fibrant simplicial sets and $f:X\rightarrow Y$ a fibration. If for any two vertices $x,x'$ of $X$, the homotopy ...
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0answers
65 views

The fiber is a homotopy fiber

The problem comes from a paper of Bertrand Toën, Homotopical Algebraic Geometry II, Appendix A, Prop A.0.3. Let $M $ be a model category and $C $ be a full subcategory of the category of weak ...
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1answer
39 views

geometric realization of a map is a strong deformation retract

A small category $\mathcal C$ having $O$ as its set of objects is called free if there exists a set $S$ of non-identity maps in $\mathcal C$ such that every non-identity map in $\mathcal C$ can ...
2
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1answer
199 views

Is the map $\Lambda^n_n \sqcup_{\Delta^{\{n-1,n\}}} J \to \Delta^n \sqcup_{\Delta^{\{n-1,n\}}} J$ left anodyne?

This question is about objects studied in chapter 2 of Lurie's Higher Topos Theory. Let $p : X \to S$ be a left fibration, and let $e \in X_1$ be an edge of $X$ such that $p(e)$ is an equivalence in $...