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
4
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Kan–Thurston theorem and R-completion
A corollary of the Kan and Thurston theorem states that the space $X$ (path connected) can be recovered, up to homotopy, by applying the fiber-wise $\mathbb{Z}$-completion functor of Bousfield–Kan (...
2
votes
0
answers
141
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Chain-level representability of simplicial cohomology
There's a simple construction for a classifying space $K(G,1)$ of a finite group $G$ as an infinite simplicial complex consisting of one $i$-simplex for each possible simplicial $1$-cocycle restricted ...
5
votes
0
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121
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When is the fat join a monoidal structure?
This question is about the following general construction.
Definition:
Let $(\mathcal C, \otimes)$ be a cocomplete, monoidal biclosed category whose unit $\ast$ is terminal. Let $I$ be an "...
22
votes
2
answers
2k
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Why are quasi-categories better than simplicial categories?
There are many models for $(\infty,1)$-categories: simplicial categories, Segal categories, complete Segal spaces, and quasi-categories.
Doubtlessly the model most used to do higher category theory in ...
7
votes
1
answer
326
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How do the various homotopy 2-categories compare?
There are various models of $\infty$-categories floating around, so there are as many models of the associated homotopy 1- and 2-categories. Because the relations between the former are worked out in ...
2
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0
answers
215
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Formally étale maps of animated $k$-algebras
In Lurie's DAG, he defines what it means for a natural transformation $T:\mathcal{F}\to\mathcal{F}'$ of functors $\mathcal{F},\mathcal{F}':\mathcal{SCR}\to\mathcal{S}$ to be formally étale. Namely, it ...
2
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0
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77
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Is there any research about secant varieties by using homotopical algebra or simplicial methods?
Let $A$ be a finitely generated $\mathbb{C}$-algebra with $proj A$ is a projective variety $X \subset \mathbb{P}^n$. The join $J(X,X)$ can be represented by $proj (A\otimes_kA)$ and also we can ...
2
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0
answers
90
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Reference request - Two notions of suspension /loop space agree for simplicial objects
In Goerss-Jardine they point out that there are two reasonable definitions of the (reduced) suspension of a simplicial set. One is the smash product with $S^1$ and one is the join with $S^0$, the &...
5
votes
1
answer
239
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Example of a non-$\infty$-category whose homotopy category is a groupoid
What is an example of a simplicial set $S$ such that its homotopy category $hS$ is a groupoid, but such that $S$ is not an $\infty$-category?
I know that if $S$ is an $\infty$-category, then $S$ is a ...
4
votes
0
answers
98
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Simplicial spaces and reflexive coequalisers
Let $X_\bullet$ be a simplicial space. Consider the reflexive coequaliser of $X_1\rightrightarrows X_0$, which we call $X$. Then we clearly have a map $\varphi\colon |X_\bullet|\to X$, where $|X_\...
5
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1
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310
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The localization of the span category
Suppose one has a model category $C$ with its class of weak equivalences $W$. It is possible to form a separate homotopical category $(C_{\operatorname{span}},W_{\operatorname{span}})$ which has ...
2
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1
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Explicitly calculating the homotopy fiber of a 3-cocycle in the category of simplicial sets
I am stuck at a critical step in my master's thesis. If someome can help me out here, I will give appropriate credit.
We know that the data of a 2-group $G$ can be given by a group $\tau_0(G)$ and a 3-...
11
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2
answers
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Algebraic topology and homotopy theory with simplicial sets instead of topological spaces
To quote Kerodon:
In fact, it is possible to develop the theory of algebraic topology in entirely combinatorial terms, using simplicial sets as surrogates for topological spaces.
A similar quote can ...
4
votes
1
answer
69
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Simplicial polytope with regular cones
Let $P$ be a convex simplicial polytope in $\mathbb{R}^n$. Can we find a convex simplicial polytope $P_0$ in $\mathbb{R}^n$ combinatorially equivalent to $P$, satisfying the following condition: The ...
5
votes
1
answer
396
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Geometric realisation of smooth $\infty$-stacks
Let $Sh^\infty(\mathsf{Man})$ denote the $\infty$-category of sheaves of $\infty$-groupoids over the site $\mathsf{Man}$ of smooth manifolds (if you prefer, that's the model category of simplicial ...
4
votes
1
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224
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Kan Complexes, proof of extension of a map to a product
I'm reading a book about Kan Complexes in Simplicial Homotopy Theory (Curtis), and came across this theorem at the beginning : if $f : \Delta[n] \times \Lambda^k[m] \to K$ is a simplicial map and $K$ ...
1
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1
answer
331
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Proving that the simplex category is generated by the face and generacy maps
Note: This is not intended to be a research level question, but concerns graduate level material.
Theorem. The opposite $\Delta^\mathrm{op}$ of the simplex category $\Delta^\mathrm{op}$ (as usually ...
3
votes
1
answer
280
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Can homotopy limits of simplicial sheaves be calculated (correctly) using sheaves of Kan complexes?
$\DeclareMathOperator\holim{holim}$
Let $sSh$ be the category of simplicial sheaves on some site (I like using the psychological crutch of the site having enough points; further, to clarify a bit, &...
4
votes
0
answers
110
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Why is this class of right-anodyne maps closed under pushouts?
Let $S$ be the class of all right-anodyne maps $r$ such that the pullback of $r$ along any left fibration is again right-anodyne. According to Land's book on $\infty$-categories (specifically the ...
2
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0
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232
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Why is $\operatorname{Hom}^R_{\mathcal{C}}(X, Y )$ the fiber of $\mathcal{C}_{\backslash Y} \to \mathcal{C}$ (Lurie's HTT)
Reading Jacob Lurie's Higher Topos Theory I not understand the proof of the "only if" part in Proposition 1.2.12.4. It states
Proposition 1.2.12.4. Let $\mathcal{C}$ be an $\infty$-category ...
6
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2
answers
601
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Simplicial set construction of the classifying space
What would be the best book, article, or otherwise to reference for the specific construction of the classifying space for a discrete group $G$ which goes as follows?:
Regard $G$ as a category with ...
0
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0
answers
141
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A fibration is a map which has the right lifting property with respect to injections that are weak equivalences
As mentioned in Motivic Homotopy Theory, an alternative criterion for $X\rightarrow Y$ to be a fibration is that the diagram
$$\require{AMScd}
\begin{CD}
A @>>> X;\\
@VV{i}V @VVV \\
B @>&...
7
votes
1
answer
533
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Is $\oplus$ the only monoidal structure on the simplex category?
Simplicial sets are presheaves on the simplex category $\Delta$, while augmented simplicial sets are presheaves on $\Delta_+$, the augmented simplex category. Because Day convolution allows us to ...
1
vote
0
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369
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History of simplicial complex
It is easy to find the definition of a simplicial complex:
https://en.wikipedia.org/wiki/Simplicial_complex
I am interested in the history and first occurrences of the concept.
When did people start ...
12
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3
answers
504
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Small simplicial set models for BG
Let $F$ be a finite group.
Is there a model for $BF$ as a simplicial set such that the number of nondegenerate $n$-simplices grows at most polynomially?
For example the Bar construction has the ...
4
votes
0
answers
210
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A notion of "generalized nerve" of categories enriched over a presheaf
Let $\mathcal{C}$ be a small category, $p : \mathcal{C} \to \mathsf{Cat}$ a functor, and $\mathsf{P} = \mathrm{PSh}(\mathcal{C})$ the category of presheaves over $\mathcal{C}$ valued in sets. The ...
8
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0
answers
268
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(Homotopy) inverse limits of towers of spaces or simplicial sets - reference request
Suppose we have a tower of Kan fibrations between Kan complexes:
$$ 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 ...
5
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0
answers
100
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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 ...
0
votes
1
answer
229
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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}$ ...
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 ...
1
vote
0
answers
58
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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 $\...
2
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1
answer
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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
847
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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 ...
9
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0
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268
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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^\...
2
votes
1
answer
164
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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 ...
6
votes
1
answer
166
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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 ...
2
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0
answers
138
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$\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
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0
answers
244
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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 ...
2
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0
answers
76
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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/...
2
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0
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63
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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....
12
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1
answer
802
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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 ...
9
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2
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323
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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 ...
3
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0
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262
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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}$$
by compositing the ...
6
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0
answers
187
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Base-change for simplicial spaces
Base-change for simplicial spaces
Let us say that a map of simplicial spaces $X_* \to Y_*$ is a base-change if for all $n$ the canonical map
$$
X_n \to (X_0)^{n+1} \times_{(Y_0)^{n+1}}^h Y_n
$$
is ...
6
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1
answer
1k
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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
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0
answers
109
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"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
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0
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111
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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
550
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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
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0
answers
127
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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
326
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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} \...