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.
775
questions
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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 ...
2
votes
1
answer
276
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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 ...
6
votes
1
answer
373
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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 ...
5
votes
2
answers
423
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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 ...
12
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1
answer
290
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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 ...
4
votes
1
answer
270
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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 ...
5
votes
1
answer
132
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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 ...
4
votes
1
answer
212
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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 ...
7
votes
0
answers
167
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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 ...
5
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0
answers
297
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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 ...
3
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0
answers
119
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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 (...
2
votes
0
answers
115
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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 ?
...
7
votes
0
answers
190
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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 ...
2
votes
1
answer
115
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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-...
4
votes
1
answer
207
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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 ...
3
votes
0
answers
64
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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 ...
4
votes
1
answer
342
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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 ...
2
votes
1
answer
241
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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/...
4
votes
1
answer
342
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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 ...
4
votes
2
answers
342
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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 ...
3
votes
0
answers
157
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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\...
7
votes
1
answer
160
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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 ...
2
votes
1
answer
283
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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 ...
7
votes
1
answer
400
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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 ...
4
votes
1
answer
242
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($1$-)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 ...
1
vote
0
answers
187
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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 ...
5
votes
1
answer
443
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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
199
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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, ...
4
votes
0
answers
367
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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 ...
5
votes
0
answers
119
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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 ...
0
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0
answers
56
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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 ...
5
votes
0
answers
116
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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 ...
3
votes
1
answer
97
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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 ...
1
vote
1
answer
347
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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 ...
10
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0
answers
282
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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 $\Omega^n (-) := map((I^n, \partial I^n), (-, *))$...
6
votes
0
answers
126
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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 ...
8
votes
2
answers
801
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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 ...
6
votes
3
answers
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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: $X \mapsto \pi_{\leq 1}(X)$, ...
7
votes
1
answer
377
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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 ...
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:
$...
1
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0
answers
144
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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 $...
5
votes
1
answer
125
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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 ...
2
votes
0
answers
78
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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 ...
1
vote
0
answers
69
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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_* \...
1
vote
1
answer
171
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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....
4
votes
1
answer
119
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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 ...
2
votes
0
answers
156
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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?
3
votes
1
answer
87
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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 $(0,0,\ldots,0)$ ...
3
votes
0
answers
76
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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) \...
6
votes
1
answer
421
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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 ...