# Questions tagged [simplicial-stuff]

For questions about simplicial sets, simplicial (co)algebras and simplicial objects in other categories; geometric realization, Dold-Kan correspondence, simplicial resolutions etc.

599
questions

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### Computation on homotopy colimit cocomplete triangulated categories

I have a couple of questions about dealing with homotopy (co)limits cocomplete triangulated categories.
Question I:The first one concerns a comment by Peter Arndt in this discussion about derived ...

**7**

votes

**1**answer

118 views

### Groupoid completion of a topological category vs its homotopy category?

Given a category $\mathcal{C}$ enriched in spaces, we can take the nerve (a simplicial space) and then geometric realization to get a space $B\mathcal{C}$. If we view spaces as $\infty$-groupoids, ...

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155 views

### Etale sites for stacks

Let $X$ be an algebraic stack, let $U\to X$ be a smooth cover by an algebraic space. In this setting, we have the big étale site of $X$ (if $X$ is a stack over a scheme $S$, this is the restriction of ...

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92 views

### The homotopy type of the simplicial space obtained by free adding degeneracies to a semi-simplicial space

Let $\text{sTop},\text{ssTop}$ denote the categories of simplicial, semi-simplicial spaces respectively. There is a functor $E:\text{ssTop}\rightarrow \text{sTop}$ that is left adjoint to the ...

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37 views

### Questions about a structure related to simplicial complexes

While researching some superficially unrelated theory, a structure similar to the one described below presented itself to me. I'm having trouble with identifying what's the structure name. It seems to ...

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69 views

### Simplicial matrices and the nerves of weak n-categories II, III, and IV

Duskin introduced his nerve functor (see the nLab or Kerodon) in the paper
Duskin, John W. Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories. [Link].
While three ...

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**0**answers

60 views

### Multi-simplicial generalization of $\Gamma$-spaces

Is there a generalization of Segal's theorem that the inclusion of $X_1$ into $\Omega|X_*|$ is a weak equivalence for a $\Gamma$-space $X_*$ if $X_1$ is group like? Specifically, I am looking for a ...

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vote

**1**answer

203 views

### How do you prove that the category of weak equivalences of sSet is accessible?

I am trying to prove that the category of simplicial sets is a combinatorial model category by using Proposition A.2.6.15 of Lurie's book, and this requires proving that the weak equivalences are ...

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209 views

### A certain semi-simplicial space

I would like to understand whether the following construction has been studied. Let me model the pointed sphere as $S^n = I^n / \partial I^n$, and let $\Omega^n (-) := map((I^n, \partial I^n), (-, *))$...

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87 views

### Any comparison between the category of cubes and its opposite?

To model topological spaces combinatorially, one can use simplicial sets -- or cubical sets. Simplicial sets are defined as presheaves on the simplex category $\Delta$, the category of non-empty ...

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votes

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126 views

### Reference for homotopy colimit = total complex

I'm looking for a reference for the following fact:
take a simplicial chain complex $ X:\Delta^{op}\to Ch_{\geq 0}(\mathcal A)$ for $\mathcal A$ a nice abelian category (say, cocomplete with enough ...

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399 views

### What is the geometric realization of the the nerve of a fundamental groupoid of a space?

It can be easily seen that there exists a functor $F:Top \rightarrow Grpd$ from the category of topological spaces to the category of groupoids defined as follows:
Obj: $X \mapsto \pi_{\leq 1}(X)$, ...

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157 views

### The $\infty$-category of natural transformations as an end

Let $\mathcal{C}$ be an $\infty$-category viewed as a fibrant scaled simplicial set with all 2-simplices thin and let $\mathfrak{C}\!at_{\infty}$ be the $\infty$-bicategory of $\infty$-categories. A ...

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votes

**1**answer

301 views

### What is the notion of a group object and its action in a 2-category?

It is well known that a group object in a category $C$ (with terminal object $1$ and such that any two objects of $C$ have a product) is defined as an object $G$ in $C$ with the following morphisms:
$...

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86 views

### What is the meaning of lifting property? and some question in $\infty$-category

When I learn model category, the important compute tool is the lifting property between $(\operatorname{Cof}, \operatorname{Fib} \cap W)$ and $(\operatorname{Cof} \cap W, \operatorname{Fib}) $, where $...

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104 views

### Considering each half of factorization of weak equivalence separately

I have been working through the proof of Theorem 3.4.1 in the article https://arxiv.org/pdf/1211.2851.pdf (pages 35-6), but there is one technical detail still unclear to me.
Specifically, we have ...

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votes

**0**answers

68 views

### Pro-trivial cosimplicial tower of spaces

Let $\{X^\bullet_s\}$ be a cosimplicial tower of spaces. In other words, for each fixed "tower degree" n, we have a (let's assume Reedy fibrant) cosimplicial space $X^\bullet_n$, and for each fixed ...

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50 views

### Simplicial differential graded algebra and a filtration

Let $A$ be a simplicial differential algebra, i.e. for each $n \in \mathbb{N}$ a differential graded algebra $(A_n,d_n)$ and for each weakly increasing map $f \colon [n] \to [m]$ a morphism $f_* \...

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vote

**1**answer

146 views

### Showing that a certain simplicial set has levelwise small cardinality

My question concerns Section 2 of the article "The Simplicial Model of Univalent Foundations (after Voevodsky)" (https://arxiv.org/pdf/1211.2851.pdf).
Let $\alpha$ be a strongly inaccessible cardinal....

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votes

**1**answer

72 views

### On equivalences of cartesian fibrations

Let $p:X^{\natural} \to S$, $q:Y^{\natural} \to S$ be two cartesian fibrations of simplicial sets (where the marking is given by cartesian edges) and assume that we are given an equivalence of ...

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112 views

### Is every simplicial spectrum equivalent to an abelian group simplicial spectrum?

I wonder if every simplicial $S^1$-spectrum stable equivalent to an abelian group simplicial $S^1$-spectrum? It seems that I could use the stable Dold-Kan on the heart and use Postnikov towers?

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**1**answer

68 views

### Representing simplicial homotopy classes by empty cubes

I am looking for references concerning the following facts, which I believe to be true:
In a Kan complex $K$, can I use "empty cubes" $\partial(\Delta^1)^{(n+1)}$ with the vertex $(0,0,\ldots,0)$ ...

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51 views

### A name in literature for a certain kind of 2-categories

Let $tr_2: \mathrm{sSet} \to \mathrm{sSet}_{\le 2} $ be the 2-truncation functor.
Let $C$ be a 2-truncated simplicial set such that every horn $tr_2( \Lambda^2_1) \to C$ extends to $tr_2(\Delta_2) \...

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**1**answer

201 views

### Understanding the adjunction $\mathfrak{C}:\mathbf{Set}^{\Delta^{op}}\rightleftharpoons \mathbf{Cat}_\Delta:\mathcal{N}$

Let $\mathbf{Cat}_\Delta$ denote the category of simplicially enriched categories (ie simplicial objects in $\mathbf{Cat}$ with all face and degeneracy maps bijective on objects). There is a canonical ...

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**1**answer

394 views

### Classifying space BG and contractable space EG

This question is probably not research level that's why I asked it previously on MSE a week ago. Unfortunately it doesn't get much attention there and I thought I would try it here.
Choose a ...

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**0**answers

95 views

### Morphisms of hammocks in the simplicial localization

Let $\mathcal{C}$ be a category together with a wide subcategory $\mathcal{W} \subset \mathcal{C}$.
In Calculating Simplicial Localizations by Dwyer and Kan, a morphism of hammocks is defined ...

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240 views

### Computations using “Stover's spectral sequence”

In this article from 1990, Stover describes a specral sequence which converges to the higher homotopy groups of the homotopy colimit of a diagram $\underline{X}$ of topological spaces.
The second ...

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**1**answer

104 views

### Symetrical simplex category

The simplex category has for objects totally ordered sets $[n]$ , and for morphisms order-preserving functions between those sets.
We can see the totally ordered set $[n]$ of size $n$ of the simplex ...

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229 views

### Generalization of familiar theorem about singular homology to general model category

I have two questions, the first one is just wether the following statement is true or not? Is there a reference for this?
The second question is maybe related, I don't know. But anyway, given $U:\...

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436 views

### Homotopy of functors

Recently I have read two different proposals for a notion of homotopy between functors, and I am curious which contexts each best lend themselves to. The first comes from Ming-Jung Lee's 1972 paper ...

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**1**answer

361 views

### Correspondences of $\infty$-categories

In Higher topos theory and Higher algebra Lurie defines (see section 2.3.1 of HTT) a correspondence between two $\infty$-categories $C$ and $D$ as being an $\infty$-category $\mathcal{M}$ over $\Delta[...

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241 views

### abelianization and homotopy

I deleted by previous questions, seems they are too vague. Let me try to ask a more precise question.
Let $f:G\rightarrow K$ a morphism of simplicial groups such that $f$ is a weak homotopy ...

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79 views

### Is the 1-simplex left homotopic to the 0-simplex?

I know they are weak equivalent (as simplicial sets). Is there a proof that they are NOT left homotopic? Here 'left homotopy' means there is a map
$$f:\triangle[1]\times\triangle[1]\to\triangle[1]$$
...

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votes

**2**answers

466 views

### About contractibility of certain categories

Let $\mathcal{C}$ be an ordinary 1-category and suppose that there exists some object $X \in \mathcal{C}$ such that the following conditions are satisfied,
(1) For every $C \in \mathcal{C}$ we have $\...

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votes

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349 views

### Category of spaces/sheaves

Consider the following category $\mathcal C$:
An object of $\mathcal C$ is a pair $(X,\mathcal F)$ where $X$ is a space and $\mathcal F$ is a sheaf on $X$.
A morphism $(X,\mathcal F)\to(Y,\mathcal G)$...

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63 views

### Face maps of cosimplicial abelian groups under Dold-Kan correspondence

This might be a stupid question, but I feel really confused how the cosimplicial Dold-Kan works explicitly...
Recall
Let $$C_\bullet:=\dots\to C_2\to C_1\to C_0$$ be a chain complex of abelian ...

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votes

**1**answer

106 views

### Cofibrant simplicial categories

Reading about simplicial categories, and in particular the model structure in sCat, I found various sources, among which I can mention this one or section 16.2 in Riehl's book “Categorical homotopy ...

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**1**answer

114 views

### Homotopy group of exterior powers of simplicial vector spaces

Let $C^\bullet$ be a cochain complex in positive degrees, by Dold-Kan correspondence it corresponds to a cosimplicial abelian group $D\colon \Delta\to \mathrm{Ab}$.
Let $F$ be a field, let $C^{\...

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231 views

### Pullbacks and fibers in the $\infty$-category of spaces

Suppose given a commutative diagram in the $\infty$-category of spaces, as the one depicted below, where all but the bottom-right squares are pullbacks. Is it true that $H$ is (equivalent to) the ...

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115 views

### A notion of fibration on bisimplicial sets

[I am not trained in this stuff, but have an outside research interest, so sorry if this question is standard.]
I am interested in notions of fibrations, or fibrant objects, in bisimplicial sets. In ...

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202 views

### Does $\pi_0$ commute with (homotopy) limits?

Consider in the category sSet of simplicial sets, let $X$ be a $J$-indexed diagram in sSet ($J$ a small category), and for each $j\in J$, $X_j$ is a Kan complex, then do we have
$\pi_0({\rm holim}_{...

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98 views

### Based loops objects in model categories

If I have a pointed model category then for I can define based loops objects as homotopy pullbacks:
$\require{AMScd}$
\begin{CD}
\Omega X @>>> *\\
@V V V @VV V\\
* @>>>...

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**1**answer

88 views

### Definition of hypercover for simplicial presheaves and hypercovering in $\infty$-topos

By lemma 4.9 in Dugger-Hollander-Isaksen, a hypercover is defined as an augmented simplicial object $U_\bullet\to X$ in the category of simplicial presheaves such that each $U_n$ is a coproduct of ...

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160 views

### Comparing definitions of cotangent complex

Consider the following two ways of defining the cotangent complex of a ring map $R \rightarrow A$ (Let $P^{\bullet} \rightarrow A$ be a polynomial resolution):
As the complex $\Omega^1_{P^{\bullet}/R}...

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votes

**1**answer

152 views

### Descent in the injective model structure and descent for simplicial presheaves

In Jardine's book Local Homotopy Theory P.102, a simplicial presheaf $X$ on a site $C$ is said to satisfy descent if some local injective fibrant replacement $ j : X → Z $ is a sectionwise weak
...

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**1**answer

179 views

### Does a homotopy sheaf functor commute with group completion

Let $\text{sPre}(C)$ be the category of simplicial presheaves with Grothendieck topology $\tau$. Let $\pi_n^{\tau}$ be the $\tau$-homotopy sheaves defined as follow
Does $\pi_n^{\tau}$ commute with ...

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194 views

### Which spaces are most naturally presented simplicially?

It's often a great technical convenience to know that you can "do homotopy theory" with simplicial sets. But if you really get down to it geometrically, it can be awkward.
In general, if I have a CW ...

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votes

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91 views

### Is transfinite composition of weak equivalences of simplicial presheaves a weak equivalence?

In a left Bousfield localization of the projective model structure on the category of simplicial presheaves, what is the condition that transfinite composition preserves weak equivalences? How about ...

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**1**answer

145 views

### Rigidification of marked simplicial sets

It is well known that there exists a Quillen equivalence,
$$\mathfrak{C}: Set_{\Delta} \rightleftarrows Cat_{\Delta}: N, \enspace \mathfrak{C} \dashv N$$
between Joyal's model structure on ...

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213 views

### When is the localization of all hypercovers equivalent to that of Čech covers?

In Theorem A.6 of Dugger's paper, it is shown that a few localizations are equivalent to the localization of Čech covers.
The Nisnevich localization
at all hypercovers is equivalent to the ...