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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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 “...
Dmitri Pavlov's user avatar
98 votes
6 answers
14k views

Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?

My (limited) understanding is that simplicial methods tend to be used whenever you want some kind of nontrivial homotopy theory -- for instance, to get a nice model structure, you use simplicial sets ...
Akhil Mathew's user avatar
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24 votes
1 answer
933 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 ...
Harry Gindi's user avatar
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8 votes
2 answers
805 views

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 ...
Maxime Ramzi's user avatar
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1 vote
1 answer
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Defining homotopy via endofunctors of a simplicial category

$\newcommand\sSets{\text{sSets}}\newcommand\sing{\text{sing}}\DeclareMathOperator\Hom{Hom}\newcommand\Top{\text{Top}}$I am looking for a reference describing the notions of homotopy and ...
user420620's user avatar
41 votes
7 answers
4k views

Simplicial objects

How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a [...
Kevin H. Lin's user avatar
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22 votes
3 answers
2k views

Necessity of hypercovers for sheaf condition for simplicial sheaves

I'm trying to understand where the definition of simplicial sheaf on a space/site comes from. For a presheaf $F$ of sets on a topological space $X$, the sheaf condition can be viewed as saying that ...
dhagbert's user avatar
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21 votes
2 answers
2k views

When is a topological space the homotopy colimit of an open covering?

Suppose that $X$ is a topological space and $\left(U_i \to X\right)$ is an open cover. We can associate to it the Cech diagram of this cover $$C_U:\Delta^{op} \to Top.$$ I know that for many good ...
David Carchedi's user avatar
21 votes
3 answers
2k views

Why is Kan's $Ex^\infty$ functor useful?

I've always heard that Kan's $Ex^\infty$ functor has important theoretical applications, but the only one I know is to show that the Kan-Quillen model structure is right proper. What else is it useful ...
Tim Campion's user avatar
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19 votes
4 answers
3k views

What are the fibrant objects in the injective model structure?

If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial ...
Chris Schommer-Pries's user avatar
19 votes
1 answer
784 views

Is there an analog of Kan's $Ex^\infty$ functor for quasicategories?

Is there a fibrant replacement functor in the Joyal model structure which can be described non-recursively, like $Ex^\infty$ for the Quillen model structure? I believe another way to put this is to ...
Tim Campion's user avatar
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16 votes
1 answer
2k views

Why does the singular simplicial space geometrically realize to the original space?

I have seen it claimed that (for compactly generated Hausdorff spaces) the geometric realization of the singular (internal) simplicial space is homotopy equivalent to the original space. I know how to ...
Chris Schommer-Pries's user avatar
14 votes
3 answers
583 views

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\...
Tyrone's user avatar
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14 votes
1 answer
641 views

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,...
Harry Gindi's user avatar
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11 votes
2 answers
1k views

Is the geometric realization of a level-wise weak equivalence a weak equivalence?

For the purposes of this question a topological space will mean a compactly generated weak Hausdorff space, though I am actually somewhat flexible on what category of topological spaces we use. I ...
Chris Schommer-Pries's user avatar
8 votes
1 answer
154 views

Morphisms of hammocks in the simplicial localization

Let $\mathcal{C}$ be a category together with a wide subcategory $\mathcal{W} \subset \mathcal{C}$. In Calculating Simplicial Localizations by Dwyer and Kan, a morphism of hammocks is defined to ...
F.Abellan's user avatar
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8 votes
1 answer
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What's the best way to test if a sphere is a polytope? (algorithms for the Simplicial Steinitz Problem)

The problem of recognizing whether a simplicial face lattice is polytopal is sometimes called the Steinitz problem. Sturmfels and Bokowski advanced a set of methods in the late 80s to test whether ...
manifold-destiny's user avatar
6 votes
3 answers
1k views

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: $X \mapsto \pi_{\leq 1}(X)$, ...
Adittya Chaudhuri's user avatar
6 votes
3 answers
2k views

Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration

This must be an elementary question: could somebody tell me a reference for the Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration? I'm working in the category of pointed ...
Agustí Roig's user avatar
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6 votes
1 answer
1k views

Homotopy pullbacks of simplicial spaces, and Bousfield-Friedlander

Let $X_\bullet \longrightarrow Y_\bullet \longleftarrow Z_\bullet$ be a diagram of simplicial spaces (=bisimplicial sets, if you like). On p. 14-9 of these notes there is an example which shows that ...
Oscar Randal-Williams's user avatar
5 votes
1 answer
818 views

When is homotopy orbit space weakly equivalent to orbit space, other than situation of free action?

Let $M$ be a closed symmetric monoidal model category. Let $X$ be a cofibrant object (it can also be fibrant if you like) and let $\Sigma_n$ act on $X^{\otimes n}$ by permuting the factors (note that ...
David White's user avatar
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4 votes
0 answers
233 views

category of simplicial filters

I am looking for references discussing the category $Filt$ of filters (in the sense of set theory, details below), its simplicial category $sFilt$ and its full subcategory $Top\hookrightarrow sFilt$ ...
user117509's user avatar
59 votes
1 answer
32k views

If I want to study Jacob Lurie's books "Higher Topoi Theory", "Derived AG", what prerequisites should I have?

I've been told that it's important to know modern physics, Differential Geometry and Algebraic Topology for understanding higher structures. Is there any other prerequisite for understanding Lurie's ...
Chuang 's user avatar
  • 609
29 votes
4 answers
4k views

Model structure on Simplicial Sets without using topological spaces

The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are ...
J Williams's user avatar
  • 1,302
25 votes
5 answers
3k views

Testing simplicial complexes for shellability

Question Are there efficient algorithms to check if a finite simplicial complex defined in terms of its maximal facets is shellable? By efficient here I am willing to consider anything with smaller ...
Vidit Nanda's user avatar
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24 votes
4 answers
1k views

How many simplicial complexes on n vertices up to homotopy equivalence?

Fix a number $n$, and define $\gamma(n)$ to be the number of simplicial complexes on $n$ unlabeled vertices up to homotopy equivalence. It is unlikely that an explicit formula exists, but what is ...
Vidit Nanda's user avatar
  • 15.4k
20 votes
6 answers
2k views

A canonical and categorical construction for geometric realization

There is a very intimate connection between categories, simplicial sets, and topological spaces. On one hand, simplicial sets are the presheaf category on the category $\Delta$ and $\Delta$ is a ...
David Carchedi's user avatar
20 votes
2 answers
1k views

Is there a discrete Cerf theory?

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the ...
Daniel Moskovich's user avatar
20 votes
3 answers
3k views

What is the homotopy theory of categories?

I've heard that Grothendieck, in his letter "Pursuing Stacks," wanted to find alternative models for the classical homotopy category of CW complexes and continuous maps (up to homotopy), and one of ...
Akhil Mathew's user avatar
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20 votes
2 answers
3k views

What is a symmetric monoidal $(\infty,n)$-category?

This question arose from reading Jacob Lurie's "Classification of topological field theories" paper. In that paper, he uses complete $n$-fold Segal spaces as a model for $(\infty,n)$-categories, but ...
Johannes Ebert's user avatar
19 votes
1 answer
856 views

Does the category of strict $2$-categories together with Dwyer-Kan equivalences provide a model for $(\infty,1)$-categories?

The question is the title. In what follows, all $2$-categories and $2$-functors will be strict. Let $2-Cat$ denote the categories whose objects are $2$-categories and whose morphisms are $2$-functors....
Jonathan Chiche's user avatar
16 votes
2 answers
1k views

The symmetric monoidal category of finite sets

It is well-known that the (augmented) simplex category is the universal monoidal category with a monoid object. What about a commutative analogue? Consider the category $\mathsf{FinSet}$ of finite ...
Martin Brandenburg's user avatar
16 votes
2 answers
3k views

Semi-simplicial versus simplicial sets (and simplicial categories)

Hi, Let's denote by "semi-simplicial set" a simplicial set without degeneracies, i.e. a contravariant functor from the category $\Delta_{inj}$ of finite linearly ordered sets and order preserving ...
Peter Arndt's user avatar
15 votes
3 answers
2k views

Does the classification diagram localize a category with weak equivalences?

Let $(C,W)$ be a category equipped with a subcategory of weak equivalences. Its "classification diagram" or "bisimplicial nerve" $N(C,W)$ is a bisimplicial set, for which $N(C,W)_n$ is the nerve of ...
Mike Shulman's user avatar
14 votes
1 answer
401 views

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, ...
Matthias Ludewig's user avatar
14 votes
3 answers
1k views

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 ...
Florian's user avatar
  • 319
13 votes
3 answers
1k views

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 ...
Jonah Ostroff's user avatar
13 votes
2 answers
659 views

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 $\...
F.Abellan's user avatar
  • 447
12 votes
2 answers
636 views

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 ...
Ilan Barnea's user avatar
  • 1,334
12 votes
1 answer
1k views

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 ...
Ronald Bernard's user avatar
12 votes
1 answer
802 views

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 ...
Giulio Lo Monaco's user avatar
12 votes
1 answer
750 views

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 $A$. ...
Andrea Gagna's user avatar
12 votes
1 answer
687 views

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 ...
Bruno Stonek's user avatar
  • 2,914
12 votes
0 answers
880 views

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(...
Richard Hepworth's user avatar
12 votes
0 answers
834 views

$E_{\infty}$ $R$-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 ...
user40276's user avatar
  • 2,199
11 votes
4 answers
1k views

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\...
archipelago's user avatar
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11 votes
1 answer
1k views

Why does the internal singular simplicial space realize to the same thing as the discrete singular simplicial set?

There are two version of the singular simplicial space of a topological space $X$, one discrete and one internal. At least if X is nice, both of them have homotopy equivalent geometric realizations (...
Chris Schommer-Pries's user avatar
11 votes
2 answers
524 views

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 ...
Ryan Budney's user avatar
  • 42.8k
11 votes
1 answer
480 views

Which maps of simplicial sets geometrically realize to fibrations?

If $f:X\to Y$ is a Kan fibration of simplicial sets, then its geometric realization $|f| : |X|\to |Y|$ is (in some suitable convenient category of topological spaces, like compactly generated ones) a ...
Mike Shulman's user avatar
10 votes
3 answers
1k views

Combinatorial distance between simplicial complexes

Let $K_1$ and $K_2$ be two simplicial complexes. I am seeking a measure of the distance between $K_1$ and $K_2$ when viewed as combinatorial objects. What I have in mind is something like this. ...
Joseph O'Rourke's user avatar