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.
254
questions with no upvoted or accepted answers
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Is this a model for $K$-theory of a triangulated category?
The recent question Complete the following sequence: point, triangle, octahedron, . . . in a dg-category reminded me of something I wanted to clarify long time ago; most likely this is now well known ...
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Hodge star and harmonic simplicial differential forms
Is there a notion of harmonic forms and Hodge theory for Sullivan's piecewise smooth differential forms on a simplicial set?
Let me recall some background.
Hodge Theory on a Riemannian manifold
A ...
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381
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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 ...
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670
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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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438
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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 ...
13
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560
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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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395
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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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12
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254
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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 $\...
12
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885
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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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842
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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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638
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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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451
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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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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), (-, *))$...
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398
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Higher Adeles of a scheme and related topics
Let $X$ be a noetherian scheme. I will describe a construction of a simplicial ring which I think is called the Bellinson higher Adeles complex (or something similar).
Consider the augmented ...
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What suffices to check completeness in an n-fold Segal space?
Recall that a Segal space is a simplicial space $X : \Delta \to \mathrm{Spaces}$, $\bullet \mapsto X_\bullet$, which satisfies the
Segal condition: For each $j$, the map
$$ X_j \to \underbrace{...
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510
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When does a sheaf of categories represent a homotopy sheaf?
Suppose that $F$ is a sheaf of categories (on a Grothendieck site or even a topological space). By this, I mean a sheaf in the naive 1-categorical sense, so it can equivalently be viewed as a category ...
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193
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Homotopy theory of suplattices
In Quillen's monograph Homotopical algebra where he introduced the notion of model category, he showed that if $C$ is a bicomplete category with enough regular-projectives in which either (*) every ...
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230
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Applications of the simplex $2$-category and its higher dimensional cousins
The simplex category $\Delta$ has a $2$-category refinement $\Delta_2$ given by the full sub-$2$-category of the $2$-category $\mathsf{Cat}$ spanned by the ordinal categories $𝟘$, $𝟙$, $𝟚:=𝟙\star𝟘...
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Is the category of all topological spaces, including the bad ones, simplicially tensored and cotensored?
Let $\textbf{Top}$ be the category of all topological spaces, including the bad ones.
We can make $\textbf{Top}$ into a simplicially enriched category as follows:
Given topological spaces $X$ and $Y$,...
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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^\...
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475
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About Kan-Thurston theorem
The Kan-Thurston theorem obsesses me recently, which is proved by Daniel Kan and William Thurston, Every connected space has the homology of a K(π,1), Topology Vol. 15. pp. 253–258, 1976. Later, there ...
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605
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Homotopy pullbacks of simplicial sets; Joyal vs Kan-Quillen model structures
I am interested in comparison of homotopy pullback squares in the category of simplicial sets with respect to Joyal' model structure and Quillen's one.
Suppose we are given a (strict) pullback square
...
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317
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Is there a citeable source for generators and relations of simplicial sets?
Simplicial sets can be specified using generators and relations,
in complete analogy with groups, rings, etc.
More precisely, a system of generators of relations for a simplicial set
consists of a ...
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158
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Are the unwound thin realization and fat realization homotopy equivalent?
This is a question about a theorem (proposition 2) in the article--On the homotopy type of classifying spaces
Recall some definitions first:
Given a category $\mathcal{C}$ internal in $\...
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157
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Why should we regard $PL(M)$ as a simplicial group?
Let $M$ be a manifold. If $M$ is smooth, it is clear what $\text{Diff}(M)$ should be, namely it should be the set of diffeomorphisms of $M$ equipped with the topology in which a sequence of ...
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280
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Tangent space, metrics etc. on simplicial sets
Is there a way to attach some sensible notion of tangent space to a simplicial set? If yes, is it possible to transfer typical local data from differential geometry such as metrics to this setting?
...
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596
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Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes
Recall that we may define a functor $\xi:\Delta\to \operatorname{Poset}$ sending a simplex $[n]$ to the set of monotone injections $[k]\hookrightarrow [n]$ for $k\geq 0$ (effectively, $k\leq n$ as ...
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Isbell duality for simplicial sets
$\newcommand{\IsbellSpec}{\mathsf{Spec}}\newcommand{\IsbellO}{\mathsf{O}}\newcommand{\Nat}{\mathrm{Nat}}$Isbell duality sets up an adjunction (see here for a short abstract summary)
$$\mathsf{O}\dashv\...
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8
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270
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(Homotopy) inverse limits of towers of spaces or simplicial sets - reference request
Suppose we have a tower of Kan fibrations between Kan complexes:
$$ 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 ...
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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 ...
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232
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Dugger and Spivak's combinatorial proof of Joyal's isofibration theorem, a fortiori?
In what follows, let $E^n = \operatorname{cosk}_0(\Delta^n)$
Joyal's isofibration theorem says precisely
An inner fibration $p:X\to Y$ of quasi-categories is a fibration for the Joyal model ...
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169
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Local formula for the signature of $4k$-manifold
In "The Euler characteristic is the unique locally determined homotopy invariant of finite complexes" on page 61 in the penultimate paragraph, Levitt mentions that if one restricts to compact PL $4k$-...
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324
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Grothendieck - A group as a sheaf over simplicial complexes
In this blog post, Terence Tao gives the following definition of a group.
Definition. A group is (identifiable with) a (set-valued) sheaf on the category of simplicial complexes such that the ...
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Simplicial covering map
In Rezk's paper "A model for the homotopy theory of homotopy theory" numerous references to simplicial covering maps are made. It's first appearance being at the bottom of page 8. Unfortunately no ...
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Simplicial Representations of (Hyper)Graph Complexes
For graph complexes, which are families of graph [on a fixed number of vertices n] closed under the deletion of edges, there is a natural simplicial complex capturing that information. Specifically, ...
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Can cyclic and simplicial objects be related in a similar way to how the species of linear orders is the derivative of the species of cyclic orders?
The derivative of a combinatorial species $S: core(FinSet) \to core(FinSet)$ is given by $S^\prime [N] = S[N\sqcup 1]$. Intuitively, an $S^\prime$-structure is built by introducing a "hole" ...
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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 ...
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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 ...
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270
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Generalization of familiar theorem about singular homology to general model category
I have two questions, the first one is just wether the following statement is true or not? Is there a reference for this?
The second question is maybe related, I don't know. But anyway, given $U:\...
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220
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Which spaces are most naturally presented simplicially?
It's often a great technical convenience to know that you can "do homotopy theory" with simplicial sets. But if you really get down to it geometrically, it can be awkward.
In general, if I have a CW ...
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283
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Does geometric realization commute with passing to the compactly generated topology?
My question is in the title, but here is a more detailed formulation:
Let Top be the category of all topological spaces and continuous maps, and let CGTop be the subcategory of compactly generated ...
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392
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Does the fat realization of simplicial spaces commute with finite limits up to homotopy?
I'm willing to work in the category of compactly generated Hausdorff spaces.
The ordinary geometric realization of a simplicial space commutes with taking pullbacks and preserves the terminal object, ...
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320
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Homotopy pullback preserving functor
In the paper http://arxiv.org/pdf/math/0101162.pdf, the authors claim during the proof of Prop. 4.2 that a functor $F:A \to B$ which preserves fibrations and weak equivalences preserves homotopy ...
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452
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Is there a higher, "orientalish" version of geometric realisation?
Geometric realisation of simplicial sets can be roughly thought of like this:
In some category $\mathcal{C}$, we choose an object for every abstract $n$-simplex. In topological spaces, we would ...
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372
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Trying to understand straightening functor associated to a right fibration of simplicial sets
Let $p:X \to S$ be a right fibration of simplicial sets; one can roughly think of it as some sort of "functor" $S^{op} \to Set_\Delta$ (where $Set_\Delta$ denotes simplicial sets) sending $s \in S$ ...
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204
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Higher categories using just simplicial sets
Is there a definition of $(\infty, n)$-category using just simplicial sets?
This is the case for $n \leq 2$.
Is the forgetful functor from saturated $n$-trivial complicial sets to simplicial sets an ...
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128
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Mapping space between $n$-groupoids is an $n$-groupoid
Consider two simplicial sets $K$ and $L$. Their mapping space (or mapping complex) is the internal hom of simplicial sets, i.e. $\underline{\mathrm{Hom}}(K,L)$, where
$$
\underline{\mathrm{Hom}}(K,L)...
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373
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Higher Algebra, Section 2.2.2
I am reading Section 2.2.2 of Higher Algebra by Jacob Lurie. There are proofs which I cannot understand, so I need someone's help.
First, I cannot understand the proof of Lemma 2.2.7. In the proof, ...
6
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0
answers
230
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Higher Algebra, Propositions 2.3.4.5 and 2.3.4.9
I am reading the proof of Propositions 2.3.4.5 of Higher Algebra by Jacob Lurie. There is a part that I don't understand, and I need someone's help.
In the book, Lurie introduces the notion of ...
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266
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Functorial identification of the mapping spaces of the arrow category of an $\infty$-category
Let $\mathcal{C}$ be a small $\infty$-category. Using the straightening-unstraightening construction, we can define a hom-functor of $\mathcal{C}$ as a functor $\mathcal{C}(-,-):\mathcal{C}^{\mathrm{...
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