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.

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Derived Category of strictly simplicial algebraic space vs. systems of objects in the derived categories

Let $X_{\bullet}^+$ be a strictly simplicial algebraic space and for a morphism $\delta:[m]\to[n]$ in $\Delta^+$, let $\delta:X_n\to X_m$ also denote the associated map (by abuse of notation). Then ...
Nikolas Kuhn's user avatar
2 votes
1 answer
276 views

question about notation in HTT of J.Lurie

In page 27 in HTT of J.Lurie, the expression $$\text{Map}_S(X,Y):=Y^X\times_{S^X}\{\phi\}\in \text{Set}_\Delta$$ appears for simplicial set $X,Y,S$ in Warning 1.2.2.2. However, I couldn't understand ...
afdsfasdf's user avatar
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1 answer
373 views

What are some "good" examples of Kan simplicial manifolds?

According to the definition 1.1 of the paper Kan Replacement of simplicial manifolds by Chenchang Zhu https://arxiv.org/pdf/0812.4150.pdf, A Kan simplicial manifold is a simplicial manifold $X$ such ...
Adittya Chaudhuri's user avatar
5 votes
2 answers
423 views

Regular or h-regular CW-complex structure for the Poincaré homology sphere

I am looking for a regular (the characteristic maps of the cells are homeomorphisms) or h-regular (the characteristic maps of the cells are homotopy equivalences) CW-complex structure for the Poincaré ...
D1811994's user avatar
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12 votes
1 answer
290 views

Can one show corbordism invariance of the Crane-Yetter state-sum using simplicial methods / are there 'Pachner-like' moves for cobordisms?

Let $\mathcal{C}$ denote some Unitary Braided Modular Fusion Category. It is well known that the Crane-Yetter state-sum, $Z_{CY}(\bullet|\mathcal{C})$ is an oriented-cobordism invariant. In other ...
Joe's user avatar
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4 votes
1 answer
270 views

A cochain complex using degeneracy maps

In constructing singular homology for a topological space, the boundary operator for the singular chain complex is given as an alternating sum of face maps. The degeneracy maps seem to be discarded in ...
Mathemologist's user avatar
5 votes
1 answer
132 views

Why is this condition necessary for the existence of a transferred simplicial model structure?

In chapter II of Goerss and Jardine's text on simplicial homotopy theory, they give a general theorem, Theorem 6.8, for transfer of simplicial model structures across a simplicial adjunction. This ...
Doron Grossman-Naples's user avatar
4 votes
1 answer
212 views

Computing homotopy colimit of a space with free $S^1$-action

Context. I am trying to understand the argument in B.4 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799 (on p147). I am still lost. But from Maxime's helpful ...
Bryan Shih's user avatar
7 votes
0 answers
167 views

Stabilisation of crossed modules?

D. Conduché has introduced a notion of a stable crossed module ("Modules croisés généralisés de longueur 2", JPAA 34 (1984) pp155–178, doi:10.1016/0022-4049(84)90034-3, Def. 3.1). Is there a ...
Matthias Künzer's user avatar
5 votes
0 answers
297 views

Čech nerve $C(U)$ corresponds to $BG$ in same manner as a hypercover $\mathcal{H}(U)$ to

We can via the bar construction canonically associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps ...
user267839's user avatar
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does geometric realization factor through an endofunctor?

Does the functor of geometric realization of a simplicial set as a topological space, factor through an endofunctor of the category of simplicial topological spaces which does something non-trivial (...
user167192's user avatar
2 votes
0 answers
115 views

when is the pullback along the shift (decalage) morphism a direct product, in sSets

What is the meaning of the following condition on a morphism in sSets or simplicial topological spaces: a morphism becomes a direct product after the pullback along the shift(decalage) morphism ? ...
user167192's user avatar
7 votes
0 answers
190 views

2d TQFTs with values in simplicial sets and Reedy categories

Let $Cob$ be the category such that $Obj(Cob)$ is $\emptyset\sqcup\mathbb{N}$, with $n$ seen as the union of $(n+1)$ circles numbered from $0$ to $n$, morphisms are (homeomorphism classes of ...
Sergei Burkin's user avatar
2 votes
1 answer
115 views

Kan replacement of finite $\mathbb{Q}$-type simplicial set

Sorry if this question is maybe a bit basic, but it is on a rather specialized topic so I think it is more appropriate for MO than SE. Suppose that $X$ is a simplicial set that has finitely many non-...
Daniel Robert-Nicoud's user avatar
4 votes
1 answer
207 views

Higher homotopy groups of Joyal fibrant replacements of 2-coskeletal simplicial sets

Suppose $X$ is a 2-coskeletal simplicial set (meaning $X^{Δ^k}→X^{∂Δ^k}$ is an isomorphism for all $k≥3$). What is the easiest example of $X$ such that the Joyal fibrant replacement $Y$ of $X$ is not ...
Dmitri Pavlov's user avatar
3 votes
0 answers
64 views

Homotopy limits of section spaces

Let $\mathcal{U}$ be an open cover of a topological space $B$. As we see, for example in this question, the associated Cech diagram $B_{\mathcal{U}}$ constitutes a simplicial space with the weak ...
Lukas Miaskiwskyi's user avatar
4 votes
1 answer
342 views

Contractibility of the category of cosimplicial resolutions

Let $\gamma : \mathcal{C} \to \mathcal{M}$ be a functor and define a cosimplicial resoultion of $\gamma$ as a functor $\Gamma: \mathcal{C} \to \mathcal{M}^{\Delta}$ such that $\Gamma C$ is Reedy ...
display llvll's user avatar
2 votes
1 answer
241 views

What is the difference between Path $\infty$-groupoid and Smooth Fundamental $\infty$-groupoid of a smooth space?

A couple of days back I asked a question Is there a Geometric/Smooth version of Homotopy Hypothesis using the path $\infty$-Groupoid of a Smooth Space? in MO about the existence of a possible Smooth/...
Adittya Chaudhuri's user avatar
4 votes
1 answer
342 views

Homotopy descent and cohomology

I've been reading "Structured Brown representability via concordance" by D.Pavlov (https://dmitripavlov.org/concordance.pdf) and I'm strugeling with a point and was wondering if someone ...
curious math guy's user avatar
4 votes
2 answers
342 views

Non-degenerate simplexes in a Kan complex

I have the following question on simplicial sets: a non-constant Kan complex has a non-degenerate simplex in every sufficiently large simplicial degree? It's Exercise 8.2.3 (p. 262) of Charles ...
Lao-tzu's user avatar
  • 1,856
3 votes
0 answers
157 views

Homotopy Kan extensions, formally coherent functors and derived Schlessinger criterion

Let $k$ be a finite field. Denote by $discArt_k$ the category of Artinian rings with residue field $k$ and $Art_k$ the category of Artinian simplicial rings. Consider a functor $\mathcal{F}:disArt_k\...
curious math guy's user avatar
7 votes
1 answer
160 views

Locally minimal simplicial categories

Given a (fibrant) simplicially enriched category $\mathcal{C}$, I'm interested in the possibility of replacing it with a weakly equivalent one (in Bergner model structure) such that all the mapping ...
Giulio Lo Monaco's user avatar
2 votes
1 answer
283 views

Meaning of "combinatorial data"

I saw several times that often some data describing certain algebraic objects, eg the set of cells of a simplical complex or a Cech cycle of a chosen coving of a variety are called "combinatorial ...
user267839's user avatar
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7 votes
1 answer
400 views

Visualize (co)sketeton of a simplicial set (geometrical intuition)

I want to understand if there is an intuition approchable with most possible 'elementary geometrical' knowledge for $n$-(co)skeleta of simplicial sets? Formally sketleton & coskeleton functions ...
user267839's user avatar
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4 votes
1 answer
242 views

($1$-)pullbacks of Kan complexes

Ultimately, I'm trying to figure out whether or not the full subcategory in $\mathbf{sSet}$ spanned by Kan complexes is finitely complete (as a $1$-category). Since fibrations are stable under ...
Zach Goldthorpe's user avatar
1 vote
0 answers
187 views

Schlessinger criterion and finiteness of tangent space

Schlessinger's criterion allows us to study whether or not a functor $\mathcal{F}:C_{\Lambda}\rightarrow \text{Set}$ from the category of local Artin $\Lambda$-algebras to sets is representable. One ...
curious math guy's user avatar
5 votes
1 answer
443 views

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 ...
user267839's user avatar
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7 votes
1 answer
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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, ...
Chris Schommer-Pries's user avatar
4 votes
0 answers
367 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 ...
V.A. Vicoji's user avatar
5 votes
0 answers
119 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 ...
user159744's user avatar
0 votes
0 answers
56 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 ...
Łukasz Lew's user avatar
5 votes
0 answers
116 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 ...
Emily's user avatar
  • 10.3k
3 votes
1 answer
97 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 ...
Connor Malin's user avatar
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1 vote
1 answer
347 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 ...
Sebastian H. Martensen's user avatar
10 votes
0 answers
282 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 $\Omega^n (-) := map((I^n, \partial I^n), (-, *))$...
Oscar Randal-Williams's user avatar
6 votes
0 answers
126 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 ...
Dasha Poliakova's user avatar
8 votes
2 answers
801 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 ...
Maxime Ramzi's user avatar
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6 votes
3 answers
1k 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: $X \mapsto \pi_{\leq 1}(X)$, ...
Adittya Chaudhuri's user avatar
7 votes
1 answer
377 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 ...
F.Abellan's user avatar
  • 447
4 votes
1 answer
519 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: $...
Adittya Chaudhuri's user avatar
1 vote
0 answers
144 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 $...
Runlei Xiao's user avatar
5 votes
1 answer
125 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 ...
CuriousKid7's user avatar
2 votes
0 answers
78 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 ...
guest's user avatar
  • 21
1 vote
0 answers
69 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_* \...
Roberto Pagaria's user avatar
1 vote
1 answer
171 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....
CuriousKid7's user avatar
4 votes
1 answer
119 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 ...
F.Abellan's user avatar
  • 447
2 votes
0 answers
156 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?
Nanjun Yang's user avatar
3 votes
1 answer
87 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 $(0,0,\ldots,0)$ ...
Daniel Robert-Nicoud's user avatar
3 votes
0 answers
76 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) \...
Andrea Marino's user avatar
6 votes
1 answer
421 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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