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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Is there an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology?
There are numerous expositions of simplicial homology in the literature.
Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes.
Hatcher in “Algebraic ...
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Asymptotics for the number of triangulations of a manifold M
In Gromov's talk at the Clay Math Research from 23:23 to 25:55 Gromov says (slightly paraphrased)
I want to emphasize a problem which
comes from mathematical physics which
is unsolved which is ...
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4
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Is there any generalization of the Dold-Kan correspondence?
The Dold-Kan correspondence gives an equivalence between simplicial abelian groups and chain complexes of abelian groups supported on negative degrees. It actually works for any abelian category. I'm ...
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What are the endofunctors on the simplex category?
Is there a 'classification' of the endofunctors F: Δ --> Δ where Δ denotes the simplex category with objects [n] and the weakly monotone maps [m] --> [n] as morphisms (Actually, I ...
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Extending Kan fibrations, without using minimal fibrations
$\require{AMScd}$One thing that needs to be checked to give an interpretation of type theory in simplicial sets (as in Kapulkin-Lumsdaine) is that "the base of the universal fibration is fibrant". ...
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What are normalized singular chains good for?
One of the common definitions of homology using the singular chains, i.e. maps from the simplex into your space. The free abelian group on these can be made into a chain complex and one can take the ...
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The weak equivalences in the covariant model structure
Let $S$ be a simplicial set. Recall that there is a model structure, called the covariant model structure (see HTT ch. 2 and this question), on $\mathbf{SSet}/S$ such that:
The cofibrations are the ...
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Is the model category of Complete Segal Spaces right proper?
Well, the title is self-explaining, I guess - I am referring to the complete Segal space model structure of Theorem 7.2 in Rezk's article "A model for the homotopy theory of homotopy theories".
Has ...
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Is every connected space equivalent to some B(Aut(X))?
Given a connected space $B$, is there always some space $X$ with $B \simeq \mathbf{B}(\mathrm{Aut}(X))$?
Here by space I mean simplicial set, by $\mathrm{Aut}(X)$ I mean the simplicial monoid of auto-...
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Strøm model structures on the category of simplicial sets
Let $X,Y$ be simplicial sets. A simplicial homotopy is a simplicial map of the form $h:X\times\Delta^1\rightarrow Y$. There are two distinguished maps
$$
in_0:X\cong X\times\Delta^0\xrightarrow{1\...
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What is the coskeleton tower of a quasi-category?
I was giving a talk in a seminar, and I mistakenly said that the coskeleton tower of a quasi-category was its Postnikov tower. Someone corrected me, but a discussion then ensued about what, precisely,...
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Realisation of maps between spheres by simplicial maps
Let $K^n_0 := \partial \Delta_{n+1}$ the simplicial set obtained by removing the $(n+1)$-simplex from the standard simplex. This gives a simplicial decomposition of the sphere $S^n$. More generally, ...
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Weak complicial sets: Are the morphisms too strict?
In Verity's first paper on weak complicial sets, he shows that every strict complicial set is a weak complicial set. He also showed in an earlier paper that the full subcategory of stratified ...
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Understand Cech Cohomology
I am currently trying to understand Cech cohomology. Five questions arised and I would be glad for help. In what follows $X$ is a topological space.
I really like Dugger's and Isaksen's paper "...
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The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?
Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying ...
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Simplicial set of permutations
Let $S_k$ be the set of all permutations of $k+1$ elements $0,1,...,k$. introduce boundary maps $d_i : S_k \rightarrow S_{k-1}$ by deleting from permutation $\eta$ element $\eta(i)$ and monotone ...
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Triangulations of polyhedra
A topologist came to me with this question, but everything I think should work doesn't.
How many triangulations are there of a polyhedron with n vertices?
By a "triangulation" of a polyhedron P we ...
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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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Multisimplicial geometric realization
Does anyone know a reference or proof for the following? Let $k\geq 1$ and let $X$ be a space. There is a $k$-fold multisimplicial set whose simplices in degree $n_1,\ldots,n_k$ are the maps $\Delta^...
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On Lemma 5.5.16 of Cisinski's "Higher Categories and Homotopical Algebra"
I have a question regarding Section 5 of Cisinski's
"Higher Categories and Homotopical Algebra".
Let us write $\mathbf{sSet}$ and $\mathbf{bisSet}$ for the categories of
simplicial sets and ...
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Group cohomology without G-modules (a.k.a. what does this bar construction compute?)
Without any prior exposure to the cohomology of groups, one might naively proceed by replacing a group by a sort of resolution.
For instance, let's take $G = \mathbb{Z}^2$, and "resolve":
$$ 0 \to \...
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Lemma 2.1.1.4 in Lurie's HTT
I have encountered a problem in understanding Lurie's proof of the following fact:
"Given a left fibration between simplicial sets $q:X \to S$, there exists a functor $$ho(S) \to Ho(sSet)$$ which is ...
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Higher Algebra, Theorem 2.4.3.18 and Remark 2.4.3.6
In his book Higher Algebra, Lurie introduces the notion of generalized $\infty$-operads ($\S$2.3.2). Roughly speaking, a generalized $\infty$-operad is a "family" of $\infty$-operads ...
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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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History of classifying spaces
Where did the idea and formal definition of the "classifying space of a (small) category" first appear?
Added: As Andy Putman noted below, the "classical" early reference for this is G. Segal's ...
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Is the simplicial completion of a localizer always a bousfield localization of the injective model structure?
Background
Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following ...
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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 ...
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Exponentiation in finite simplicial sets
A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. My question is: if $A$ and $B$ are finite simplicial sets, does this imply that the simplicial set ...
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Three questions on $\operatorname{hocolim}$
I posted this on math.stack.exchange but didn't get a helpful response, so please let me try it here.
Let $D$ be a small category and $F:D\to sSets$ a functor.
There is a bisimplicial set indicated ...
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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 ...
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Is the hom-simplicial set in the hammock localization a nerve?
Let $(C,w)$ be a relative category. Then associated to it we have its hammock localization, $L^H(C,w)$, which is a simplicially enriched category.
If $X,Y\in C$, the description of the simplicial set ...
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Is every weak $\infty$-bicategory (à la Lurie) an $\infty$-bicategory?
In Definition 4.1.1 of $(\infty,2)$-Categories and the Goodwillie Calculus I, Lurie defines a weak $\infty$-bicategory to be a scaled simplicial set that has the extension property with respect to ...
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Diameter of simplicial complex mirrored in property of Stanley-Reisner ring?
Consider a pure finite abstract simplicial complex $\Delta$. Define its diameter as the maximal distance between any two facets, i.e., between any two faces of maximal dimension $d-1$. The distance ...
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Extending a left fibration along an inner horn
Let $\Lambda^n_i \subseteq \Delta^n$ be an inner horn, and let $X \rightarrow \Lambda^n_i$ be a left fibration. Does there exist a left fibration $Y \rightarrow \Delta^n$ such that $X = Y \times_{\...
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Configuration spaces, Ran spaces, free semilattices, Vietoris spaces and power objects
These are five important constructions and I would like to know how they are related.
The $n$th unordered configuration space of a space $X$ is
$$
\operatorname{UConf}_n(X):=\{\text{embeddings of $\{...
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Tensor product of dendroidal sets: counter-examples
For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$.
...
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Can one show corbordism invariance of the Crane-Yetter state-sum using simplicial methods / are there 'Pachner-like' moves for cobordisms?
Let $\mathcal{C}$ denote some Unitary Braided Modular Fusion Category. It is well known that the Crane-Yetter state-sum, $Z_{CY}(\bullet|\mathcal{C})$ is an oriented-cobordism invariant. In other ...
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Verifying that $\epsilon^!$ is indeed the right adjoint of $\epsilon_*$ in the context of algebraic stacks
The question is about the last paragraph of Remark 12.4.3 in the book on algebraic stacks by Laumon and Moret-Bailly.
Let $S$ be a (quasi-separated) scheme and let $\mathscr{X}$ be an algebraic stack ...
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Higher homotopical information in racks and quandles
A quandle is defined to be a set $Q$ with two binary operations $\star,\bar\star\colon\ Q\times Q\to Q$ for which the following axioms hold.
Q1. a $\star$ a = a
Q2. (a $\star$ b) $\bar\star$ b = (a $\...
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Are the Alexander-Whitney and Eilenberg-Zilber maps homotopy inverse in arbitrary abelian categories?
Let $\mathcal{A}$ be a monoidal abelian category. Let $A$ and $B$ be simplicial objects in $\mathcal{A}$, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(...
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$E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras
I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral ...
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Has this chain complex associated with a simplicial complex been studied before?
I have stumbled upon a construction that has probably been noticed before, and I wonder if anyone can point me to a reference.
Suppose that $K$ is a simplicial complex. Let $P(K)$ be the free abelian ...
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What is the history of the notion of subdivision of categories?
A recent answer by Peter May prompts me to ask a question which I have been considering to ask for several months. (The reason why I have not asked it before is that it is not directly related to my ...
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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 ...
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Topological Grothendieck Construction
Let $C$ be a small category and $F\colon C^{op}\rightarrow Set$ a functor. The Grothendieck construction is the category $F\wr C$ with objects being pairs $(c,x)$ where $c$ is a object of $C$ and $x\...
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Example of a CW complex not homeomorphic to the realization of a simplicial set?
I've often heard that we can give examples of CW complexes that aren't homeomorphic to the realization of any simplicial set (although I haven't heard that there exist Kan complexes that aren't ...
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Which properties of finite simplicial sets can be computed?
A simplicial set $X$ is a a combinatorial model for a topological space $|X|$, its realization, and conversely every topological space is weakly equivalent to such a realization of a simplicial set. I ...
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Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves
Consider the global projective model category
of simplicial presheaves on some category
(the category of smooth manifolds is particularly interesting to me).
In Section 9.1 of Dugger's paper “...
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526
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Simplicial replacements in smoothing theory
As far as I can tell, ever since Milnor's Microbundles and differentiable structures (1961) paper, whenever people talk about $Diff(\mathbb R^n)$ or $PL(\mathbb R^n)$ or $Homeo(\mathbb R^n)$, they ...
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Plus construction on Simplicial Sets?
I had asked this question in Math StackExchange a few days ago, but didn't get any answers. I believe its more suitable to be asked here.
Write $\mathsf{sSet}$ for the category of simplicial sets and $...
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