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.
778
questions
7
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1
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537
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Is $\oplus$ the only monoidal structure on the simplex category?
Simplicial sets are presheaves on the simplex category $\Delta$, while augmented simplicial sets are presheaves on $\Delta_+$, the augmented simplex category. Because Day convolution allows us to ...
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 ...
7
votes
2
answers
660
views
Subdivision of simplicial sets, but not the barycentric one
Suppose $K$ and $L$ are simplicial sets. When should one consider that $K$ is a subdivision of $L$? I ask with a view to defining some notion of ‘finer’ generalising that of ’finer triangulation‘ of ...
17
votes
0
answers
381
views
Kan's simplicial formula for the Whitehead product
In his article on Simplicial Homotopy Theory (Advances in Math., 6, (1971), 107 –209) Curtis quotes a formula (on page 197) for the Whitehead and Samelson products in a simplicial group $G$. The ...
12
votes
3
answers
505
views
Small simplicial set models for BG
Let $F$ be a finite group.
Is there a model for $BF$ as a simplicial set such that the number of nondegenerate $n$-simplices grows at most polynomially?
For example the Bar construction has the ...
1
vote
0
answers
379
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History of simplicial complex
It is easy to find the definition of a simplicial complex:
https://en.wikipedia.org/wiki/Simplicial_complex
I am interested in the history and first occurrences of the concept.
When did people start ...
4
votes
0
answers
210
views
A notion of "generalized nerve" of categories enriched over a presheaf
Let $\mathcal{C}$ be a small category, $p : \mathcal{C} \to \mathsf{Cat}$ a functor, and $\mathsf{P} = \mathrm{PSh}(\mathcal{C})$ the category of presheaves over $\mathcal{C}$ valued in sets. The ...
8
votes
0
answers
270
views
(Homotopy) inverse limits of towers of spaces or simplicial sets - reference request
Suppose we have a tower of Kan fibrations between Kan complexes:
$$ X_0 \xleftarrow{f_0} X_1 \xleftarrow{f_1} X_2 \xleftarrow{f_2} \dotsb $$
From this we get a commutative diagram of topological ...
0
votes
1
answer
232
views
Simplex invariants?
Let $k=s$ be a positive definite symmetric function and a simililarity over the natural numbers such that $k(a,a) = 1 $ for all natural numbers $a$.
A similarity $s:X\times X \rightarrow \mathbb{R}$ ...
5
votes
0
answers
100
views
A simplicial analogue of a Hurewicz fibration
Let $f: X \to Y$ be a map of simplicial sets. Then there is an obvious simplicial version of demanding that $f$ be a ``Hurewicz'' fibration. Whether for every simplicial set $S$, $f$ has the lifting ...
6
votes
1
answer
104
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Non-enriched Bousfield localizations
We know that whenever we have a Bousfield localization between two simplicial model categories, this gives rise to a reflective subcategory in $\infty$-categories (or coreflective, depending on the ...
6
votes
1
answer
166
views
Join as a bifunctor
I have been reading these great notes by Charles Rezk, and one thing that has been bothering me is the join construction. To solve lifting problems in quasicategory theory we use the Leibniz ...
6
votes
0
answers
188
views
Base-change for simplicial spaces
Base-change for simplicial spaces
Let us say that a map of simplicial spaces $X_* \to Y_*$ is a base-change if for all $n$ the canonical map
$$
X_n \to (X_0)^{n+1} \times_{(Y_0)^{n+1}}^h Y_n
$$
is ...
1
vote
0
answers
58
views
Generalisation of spanning tree in simplex
This is a question I asked on Stackexchange (https://math.stackexchange.com/q/4004734/400240) but I did not receive an answer (I think it was too hard). Per someone's suggestion I put it here.
Let $\...
13
votes
0
answers
395
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 ...
2
votes
1
answer
98
views
A space $X$ is $k$-truncated iff $\text{Fun}(S,X)$ is $k$-truncated
This is probably a simple question but I can't seem to find any references for it. Call a Kan complex $X$ $k$-truncated if $\pi_n(X, x) = 0$ for all $x \in X_0$ and $n > k$.
Claim: $X$ is $k$-...
8
votes
0
answers
849
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Nonabelian variants of the Breen-Deligne resolution
The Breen-Deligne resolution is an unusual functorial resolution of an abelian group A by finite direct sums of free abelian groups of the form $\Bbb Z[A^n] = Free_{Ab}(A^n)$. It makes several ...
2
votes
1
answer
164
views
Does the monoidal structure on semisimplicial sets preserve fibrant objects?
The category of semisimplicial sets has the structure of a monoidal category by the geometric product $\otimes$, see for example Rourke and Sanderson's paper '$\Delta$-sets I: Homotopy Theory'. This ...
9
votes
0
answers
268
views
A kind of algebraic sphere?
Let $Sch$ be the 1-category of schemes. Is there a cosimplicial scheme $D^\bullet$ and a sequence of schemes $S_1, S_2, ...$ such that the geometric realization of the simplicial set $Hom_{Sch}(D^\...
6
votes
0
answers
206
views
Can representable presheaves be made injectively fibrant?
I suspect that the answer to my question is no, but let me give it a shot anyway.
If $\mathcal{A}$ is a small simplicially enriched category, then the category of simplicial presheaves $\mathsf{sSet}^...
2
votes
0
answers
139
views
$\Omega^1_{B_\bullet/A_\bullet}$ is acyclic if $A_\bullet \to B_\bullet$ is quasi-isomorphism
Let $A_\bullet \to B_\bullet$ be a quasi-isomorphism of simplicial rings in the sense of (P.62, I.3.1.7, Complexe Cotangent et Déformations I, Luc Illusie).
Then, we define the simplicial $B_\bullet$-...
2
votes
0
answers
252
views
Mapping spaces of simplicial model categories and quasicategories
Let $M$ be a simplicial model category, $M^o$ its full subcategory of bifibrant objects. The axioms of a simplicial model category guarantee that $M^o$ is enriched in Kan complexes. Thus the homotopy ...
12
votes
1
answer
815
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Modern proofs for simplicial localizations
I know that the references usually regarded as standard for simplicial localizations are the Dwyer and Kan's three articles from the 80's. I would be interested in a more modern approach to the ...
2
votes
0
answers
76
views
Left anodyne is covariant equivalence
I have a question about the proof of left anodyne between two simplicial sets over $S$, where $S$ is a simplicial set, is covariant equivalence. In the proof (HTT 2.1.4.6 or https://ncatlab.org/nlab/...
9
votes
2
answers
337
views
Simplicial spaces internally to simplicial sets
I am a master’s student with interest in topos theory and its applications (motivated by Ingo Blechschmidt’s thesis, as seems to be usual).
After finding out about some of the uses of simplicial ...
2
votes
0
answers
64
views
homotopy coherent G-action on tensor product of complexes
Let $G$ be a discrete group and $k$ a field. Suppose $C_1$ and $C_2$ are complexes over $k$ with homotopy coherent actions of $G$ in the sense of Cordier (I've been reading https://arxiv.org/pdf/1801....
6
votes
2
answers
321
views
Where to find the proof that these two version of simplicial homotopy are equivalent?
Let $f,g: X_{\bullet}\to Y_{\bullet}$ be two simplicial maps between simplicial sets. We say $f$ and $g$ are (strictly) simplicial homotopic if there exists a simplicial map
$H: X_{\bullet}\times I_{\...
3
votes
0
answers
263
views
All functors "are" left adjoints, and applications?
Throughout this thread, let us assume smallness.
All functors "are" left adjoints
Let $D \xrightarrow{F} C$ be any functor, which induces
$$ D \xrightarrow{F} \hat{C}$$
by compositing the ...
6
votes
1
answer
1k
views
Proposition 5.13 (ii) in Scholze's Perfectoid Spaces
In Proposition 5.13 (ii) in Scholze's Perfectoid Spaces, we have $R \to S$ a morphism of $\Bbb F_p$-algebras and the assumption that the relative Frobenius $\Phi_{S/R}$ induces an isomorphism $R_{(\...
3
votes
0
answers
109
views
"Boundaries" in Free Simplicial Monoids
I suspect that this has been addressed somewhere already, but I cannot find anything. Let $\hat{\Delta}$ denote simplicial sets, and $Mon\hat{\Delta}$ simplicial monoids. There is a forgetful functor $...
2
votes
0
answers
111
views
Sufficient coordinate-free condition for points being co-spheric
Question:
is there a theorem that guarantees that
$\mathcal{P}\subset\mathbb{E}^n$ is finite set of points in a Euclidean space and all radii of the $(n-1)$-spheres that are defined by the $n$-...
8
votes
1
answer
557
views
About definition of homotopy colimit of Kan and Bousfield
In Bousfield and Kan's book"Homotopy Limits, Completions and Localizations",they define homotopy direct limit for system of pointed simplicial sets(Ch XII S2 2.1 P327), while they define ...
4
votes
0
answers
127
views
Example of non reduced representable functor from simplicial rings
Let $\text{sCR}_k$ be the over category of simplicial commuative rings (so objects are maps $R \to k$), where $k$ is the simplicial field generated by a non-derived field of the same name. Let $R \in \...
6
votes
1
answer
332
views
Elementary proof of the exactness of Čech complex associated to a hypercovering ("Illusie's Conjecture")
Let $\mathcal{E}$ be a sheaf of abelian groups on a topological space (or a site). For an open covering $\mathfrak{U} = (U_i)_i$, it is well known that the augmented Čech complex $0 \to \mathcal{E} \...
3
votes
1
answer
186
views
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
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 ...
12
votes
1
answer
293
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 ...
6
votes
0
answers
991
views
The normalised cochain complex, totalisation and cosimplicial simplicial $R$-modules
Short Version
Given a cosimplicial space $X_\bullet$, what is the relationship between (co)chains on the totalisation of $X_\bullet$ and the totalisation of the cosimplicial chain complex obtained by ...
6
votes
1
answer
374
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 ...
5
votes
2
answers
425
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 ...
4
votes
1
answer
273
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 ...
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 ...
4
votes
1
answer
213
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 ...
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 ...
5
votes
0
answers
299
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 ...
3
votes
0
answers
119
views
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
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 ?
...
24
votes
1
answer
936
views
A combinatorial approximation functor sSet->qCat
Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap ...
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 ...
7
votes
1
answer
378
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 ...