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13
votes
0answers
303 views

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
votes
0answers
740 views

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 ...
10
votes
0answers
375 views

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 ...
10
votes
0answers
259 views

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 ...
8
votes
0answers
129 views

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 ...
8
votes
0answers
102 views

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 ...
8
votes
0answers
389 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 ...
7
votes
0answers
143 views

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? ...
7
votes
0answers
292 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 ...
6
votes
0answers
135 views

Uniformly sampling from the set of all simplicial maps

Let $K$ and $L$ be finite simplicial complexes that remain fixed throughout. How does one efficiently sample (according to the uniform distribution) elements from the finite set of simplicial ...
6
votes
0answers
420 views

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 ...
6
votes
0answers
631 views

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, ...
5
votes
0answers
166 views

Geometric realization of simplicial spaces and finite limits

Let $X_{\bullet}$ be a simplicial space and denote the (non-fat!) geometric realization by $\lvert X_{\bullet} \rvert$. Does this geometric realization of simplicial spaces preserve finite ...
5
votes
0answers
194 views

Left adjoints to several inclusions of homotopy simplicial model categories

The left adjoint to the inclusion $sGrp\hookrightarrow sPtSet$ of the category of simplicial groups into the category of pointed simplicial sets is homotopy equivalent to the loop suspension functor ...
5
votes
0answers
338 views

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 ...
5
votes
0answers
277 views

Expected degree of a vertex in a tetrahedralization?

Definitions I will define the degree of a vertex in a tetrahedralization to be the number of highest dimensional cells (which in this case are tetrahedra) that touch the vertex. Let $S$ be a fixed ...
4
votes
0answers
164 views

What does it mean for the $\Pi_\infty$ groupoid to be fully fibrant?

The original question I had was: If I have a sequence of simplicial spaces $$A\to B\to C$$ which is degree-wise a homotopy fibration, under which conditions is the geometric ...
4
votes
0answers
164 views

Adding morphisms to a category without changing homotopy type

I have a really tame category $C$: there are only finitely many objects $C_0$, each hom-set $C(x,y)$ for $x,y \in C_0$ has at most one element, and (aside from identity morphisms) if $C(x,y)$ is ...
4
votes
0answers
147 views

Is there a local projective model structure on simplicial sheaves? What are its fibrant objects?

Consider a site S (I am mostly interested in hypercomplete sites, e.g., the site of smooth manifolds). The category of simplicial presheaves SPSh(S) on S can be equipped with the local projective ...
4
votes
0answers
248 views

How to endow an n-fold Segal Space with a symmetric monoidal structure?

I would like to understand how I can endow an n-fold (complete) Segal Space with a symmetric monoidal structure. My question is basically the same as in this post: What is a symmetric monoidal ...
4
votes
0answers
210 views

Extending the vertex-facet correspondence from Δ to Θ

Recall that in the $n$-simplex $\Delta[n]$, we have a combinatorially crucial bijection between facets, (codimension $1$ faces) and vertices, where the $i$th face of a simplex corresponds to the full ...
4
votes
0answers
199 views

good covers and simplicial maps

Let $X$ be a paracompact topological space and choose a good cover $U_i$ of $X$. Remember that a good cover is one that consists of open subsets, such that each set $U_i$ is contractible and all ...
4
votes
0answers
184 views

Is the class of inner-anodyne morphisms right-cancellative with respect to the of the class of monomorphisms?

Recall: Given a category $A$, and two classes of morphisms $S,S'$, we say that $S$ is right-cancellative with respect to $S'$ if for any pair of maps $f\in S, g\in S'$ such that $gf$ is defined, we ...
4
votes
0answers
97 views

What is known about the number of permissible simplicial complexes given the number of k-cells?

Motivation: I am working on a problem that reduces to finding simplicial complexes given some data (details are unnecessary), but all I have managed to wrangle from my input is the number of cells of ...
4
votes
0answers
58 views

maps of oo-stacks inducing monomorphisms on all homotopy sheaves

What is known about factoring morphisms of simplicial (pre)sheaves/$\infty$-sheaves through maps that induce monomorphisms on all homotopy sheaves? Is there any useful theory for that?
3
votes
0answers
94 views

Recognition principle for 2-categories (2-groupoids)

Given a 2-category (i.e. bicategory) $C$ there is a unitary geometrical nerve whose 0-simplices are objects of $C$, 1-simplices are 1-arrows of $C$, 2-simplices are 2-commutative triangles (in certain ...
3
votes
0answers
100 views

Weak notions of Kan fibrations

I have two semi-simplicial complexes $X_\bullet$ and $Y_\bullet$, along with a simplicial map $f_\bullet: X_\bullet \to Y_\bullet$. Now $f_\bullet$ is not a Kan fibration: for an $n$-simplex in ...
3
votes
0answers
122 views

Where does the ''hyper'' go?

There is a model structure on the category $Grpd$ of groupoids such that weak equivalences are equivalences of categories, cofibrations are the injections on objects (see nlab) and there is a Quillen ...
3
votes
0answers
173 views

Simplicial chain complex with ordered simplices

Let $X$ be an abstract simplicial complex. Recall that the usual simplicial chain complex for $X$ is defined as follows. Let $C_k(X)$ be the quotient of the free abelian group on formal symbols ...
3
votes
0answers
136 views

Is the homotopy of a primitively generated Hopf algebra still primitively generated?

Let $A=\oplus A_n$ be a primitively generated graded Hopf algebra, where each $A_n$ is a simplicial group. This allows us to define the homotopy group $\pi_*(A)$. Question: is the graded Hopf algebra ...
3
votes
0answers
146 views

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 ...
2
votes
0answers
55 views

Relationship of height zero hypercovers to co-cartesian condition on cosimplicial modules

Suppose given a cosimplicial ring $R^\bullet$ and a cosimplicial module $M^\bullet$ (i.e. a cosimplicial Abelian group such that $M^n$ is an $R^n$-(left/right/bi)module). I have seen it said that ...
2
votes
0answers
143 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 ...
2
votes
0answers
126 views

Explicit Lie May structure on cosimplicial DG Lie algebras

In the paper "Homotopy Lie algebras", Schechtman and Hinich proved that any cosimplicial differential graded Lie algebra has the structure of a 'Lie May algebra'. If my understanding is right here, ...
2
votes
0answers
162 views

Which 2-coskeletal simplicial sets is the nerve of a category?

Let ${\mathrm{tr}}_2$ be the truncation functor that takes a simplicial set and restricts it to dimensions at most 2. Its right adjoint is the 2-coskeleton functor. NLab says that the nerve of a small ...
2
votes
0answers
148 views

Are connected categories with pullbacks weakly contractible?

Quillen's Theorem A says that a functor between (small) categories $f:I\rightarrow J$ induces a weak equivalence of the nerves if for each $j\in J$ the comma category $f/j$ is weakly contractible. In ...
2
votes
0answers
103 views

$W^{1,1}$ simplicial approximation.

Let $f$ be a continuous real-valued function defined on an $n$ dimesional simplex $\Sigma\subset \mathbb{R}^n $. The classical simplicial approximation scheme provides a sequence $f_k$ of piecewise ...
2
votes
0answers
186 views

On small generators of an infinity category

Suppose that $\mathcal{C}$ is an $\infty$-category with pullbacks and small coproducts. Suppose that $\mathcal{D}$ is a small subcategory for every object $C,$ the canonical map ...
2
votes
0answers
179 views

The Mayer-Viertoris exact sequence as a (Zariski) descent spectral sequence.

For certain 'spaces' $U,V$ (they are certain Henselizations of subvarieties) I would like to compute (certain etale) cohomology of $U\cup V$ in terms of the corresponding cohomology of the diagram ...
2
votes
0answers
287 views

simplicial deRham complex and model category structure

To every simplicial manifold is associated its simplicial deRham complex. Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) ...
1
vote
0answers
39 views

Prove the functor $[n]\to [n]\star [n]$ preserves inner anodyne

Let $f:\Delta\to \Delta$ be the functor given by $[n]\mapsto [n]\star[n]=[2n+1]$. We can extend $f$ cocontinuously to a functor $$f_!: SSet\to SSet$$ (that is, the left adjoint of the functor $f^*$. ...
1
vote
0answers
61 views

What is a homotopy between bisimplicial maps

I originally posted it on stackexchange, but did not get any comments or answer, so I try my luck here. I am looking for the naive notion of homotopy. For maps $f, g: A\to B$ of simplicial sets ...
1
vote
0answers
94 views

The bar construction or the quotient for monoidal category action on a category

Given a monoid $C$ acting (from the right) on a set $M$, there is a bar construction giving a simplicial set or equivalently a translation/action category, $$ N_\bullet (M\rtimes C)= \cdots M\times ...
1
vote
0answers
91 views

Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the cosimplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
1
vote
0answers
100 views

Does compactly supported cohomology make sense for cosimplicial spaces?

As far as I understand, whenever one has something (co)simplicial in spaces, one should take a sort of diagonal to reduce the study to a single space. I'm never sure though whether one preserves only ...
1
vote
0answers
101 views

Dold-Kan preserves weak equivalences and fibrations

It's well known that Dold-Kan correspondence is an isomorphism between simplicial vector spaces and non-negative chain complexes of vector spaces. Moreover, weak equivalences and fibrations are ...
1
vote
0answers
164 views

Geometric interpretation of higher simpicial homotopy groupoids.

As a geometric interpretation of the simplicial fundamental groupoid, resp. its equivalent relation I thought about something like arcwise differentiable (or arcwise linear) paths modulo "we can go ...
1
vote
0answers
103 views

Do bistellar flips in a simplicial sphere preserve facet connectivity?

Define two facets of a simplicial sphere as k-connected if there are at least k disjoint edge-vertex paths between them in their facet-ridge graphs. Do bistellar flips preserve facet k-connectivity? ...
1
vote
0answers
103 views

Simplicial morphisms from degenerate simplices

Frequently I read the following statement on morphisms of simplicial sets: Suppose $X$ and $Y$ are simplicial sets, $f: X \rightarrow Y$ is a simplicial morphism and we write $X_n$ as well as $Y_n$ ...
1
vote
0answers
113 views

Certain (pre) simplicial sets have torsion-free homology?

Suppose $X$ is a pre-simplicial set defined on a finite vertex set $\{v_1, v_2, \ldots, v_k\}$ recursively as follows: Let $X_1$ have a single vertex $\{v_1\}$. For $i > 1$, $X_i$ is obtained ...