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1
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1answer
124 views

homology of $B S^{-1} S$ computation in the proof that $+ = Q$

Let $S$ denote the category of projective (left) $R$-modules with isomorphisms for arrows. We have that $BS^{-1}S \sim B \text{GL}(R)^+ \times K_0(R)$ In proving this, in Srinivas' algebraic ...
2
votes
1answer
232 views

Path components of a monoidal category acting on homology

Let $S$ be a (small) symmetric monoidal category and $X$ a (small) category on which $S$ acts. $\pi_0(S) = \pi_0(BS)$ is naturally an abelian monoid, with $[A] + [B] := [A+B]$, where $[A]$ denotes ...
12
votes
2answers
604 views

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 ...
2
votes
2answers
262 views

Simplicial space whose all face/degeneracy maps are homotopy equivalences

I believe that the following is true, but I cannot find a proof. Let $X_\bullet$ be a simplicial topological space (I can add that my $X_\bullet$ comes from a bisimplicial set, so the spaces $X_n$ are ...
8
votes
3answers
604 views

When is the projective model structure cartesian? When is the internal hom invariant?

If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category $Fun(D,M)$ where the weak equivalences are the ...
2
votes
2answers
553 views

Simplicial path and loop spaces

I am trying to understand the relationship between the simplicial path space and loop space with the path space of a topological space, and the loop space of a topological space. I have understood ...
7
votes
2answers
295 views

realization of maps between classifying spaces of categories

The classifying space $B\mathcal{C}$ of a small category $\mathcal{C}$ is by definition the geometric realization of the nerve of $\mathcal{C}$. Now let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two ...
1
vote
1answer
219 views

Simplicial sets from bisimplicial sets, and their realisations.

From a bisimplicial space $T$, one can consider the simplicial spaces $\underline p \mapsto T_{pp} $, $\underline p \mapsto |\underline q \mapsto T_{pq} |$, and $\underline q \mapsto |\underline p ...
8
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0answers
169 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? ...
15
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3answers
1k 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 ...
1
vote
0answers
168 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 ...
9
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0answers
135 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 ...
5
votes
1answer
403 views

Is every finite subcomplex of a contractible simplicial complex contained in a finite contractible subcomplex?

The question is as in the title: Is every finite subcomplex of a contractible simplicial complex $K$ contained in a finite contractible subcomplex of $K$? What if we are allowed to take ...
5
votes
2answers
1k views

Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
12
votes
2answers
523 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 ...
2
votes
1answer
204 views

hypervolume under the square of an n-simplex

I posted this question at math.stackexchange.com, reformulated and posted again both times without much luck. I also asked a math professor at my uni who suggested I post it here. Hopefully, it is ...
7
votes
1answer
346 views

The Quillen model structure on simplicial sets as a Bousfield localization

Starting with the trivial model structure on the category of simplicial sets (that is the weak equivalences are exactly the isomorphisms and the cofibrations and fibrations are arbitrary maps), is it ...
9
votes
3answers
2k views

This is not a category. What is it?

EDIT The question was based on an error, as it turns out. In fact my example is a category (and therefore a groupoid), by Eric Wofsey's argument. I can't remember why I thought it wasn't, and I feel a ...
15
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2answers
637 views

If $X$ is a simplicial complex, is their a characterization of the links of the vertices of $X$ that is equivalent to the statement "$|X|$ is a manifold

We have a characterization when we want $|X|$ to be a PL-manifold, in particular that the links of all the vertices are themselves (PL) spheres. If we are in the category of PL- spaces then this is a ...
9
votes
1answer
649 views

Is there an algebraic “derived mapping space” construction that encompasses both Hochschild homology and loop spaces of non-simply-connected spaces?

I'm looking for directions to the literature that might contain fairly explicit constructions that might be called (the algebra of functions on) the "derived mapping space" from a simplicial set to a ...
1
vote
1answer
274 views

Relating two notions of geometric realization

Let $K$ be an abstract simplicial complex on the (finite) vertex set $V$. The geometric realization $|K|$ is typically defined (see Spanier's book for instance) as the collection of functions ...
4
votes
0answers
288 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 ...
1
vote
1answer
99 views

Does the following categorial sum preserve weak equivalences?

In Dwyer and Kan's 1980 paper on "Simplicial Localizations of Categories", they prove the following result for binary categorial sums. For a set $O$, let $O$-${\mathsf{Cat}}$ be the following ...
0
votes
0answers
60 views

Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius

I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
4
votes
1answer
248 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 ...
3
votes
1answer
276 views

Does each “prod-simplicial” regular cell complex come from a unique rooted tree?

Since about the time I asked the question, What is the precise relationship between "prodsimplicial sets" and rooted trees? I have been playing with these rooted trees and their ...
11
votes
0answers
312 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 ...
19
votes
2answers
640 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 ...
8
votes
4answers
867 views

Is there a sense in which the homotopy theory of simplicial sets is the “paradigmatic” one?

I could not come up with a better title for my question. What I am asking is this (preemptive excuses to all experts in homotopy theory for naivetes of all kinds you may find herein): the ...
3
votes
2answers
297 views

Analogues of fibrations

Recall the following analogy Serre fibrations : Kan fibrations in spaces and simplicial sets respectively, related by the singular simplices functor and geometric realisation. There are other ...
4
votes
1answer
453 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 ...
2
votes
0answers
176 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 ...
4
votes
3answers
591 views

Constructing a simplicial set homology-equivalent to a given CW complex

I would like to compute the homology of certain low dimensional CW complexes and I am hoping to take advantage of software that handles simplicial sets as input. Thus, I would like to convert a CW ...
6
votes
1answer
156 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 ...
3
votes
1answer
208 views

Explicit contraction for the universal simplicial bundle WG

For a simplicial group $G$, there is a universal bundle $WG \to \overline{W}G$ in the category of simplicial sets, detailed in for example May's book (djvu). Now $WG$ has a simple enough description ...
6
votes
1answer
198 views

Smash product of Kan complexes

Suppose $X$ and $Y$ are pointed Kan complexes. Is their smash product $X\wedge Y$ also a Kan complex? I expect the answer is probably no, but it would be nice to see a counterexample.
2
votes
1answer
342 views

On the construction of the simplicial category $\Delta$

Is a classical example that in the topos $Set$ the set of natural numbers (finite cardinals) $\mathbb{N}$ is the natural-numbers objet as in topos theory definition. Now the category $\Delta$ has for ...
9
votes
1answer
328 views

What is the precise relationship between “prodsimplicial sets” and rooted trees?

In Keven Walker's answer to the question, Cubical vs. simplicial singular homology it is written: Personally, I think it is more convenient to do singular homology with the larger collection ...
2
votes
2answers
1k views

The Area of Spherical Polygons

I am interested in finding a canonical general expression for the area of a spherical polygon in $\mathbb{S}^2$ knowing the side lengths of the polygon and a bound on the internal angles (we can ...
6
votes
2answers
422 views

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 ...
4
votes
0answers
225 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 ...
2
votes
0answers
217 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 ...
1
vote
1answer
111 views

Generalizing the internal angle of a graph in $\mathbb{E}^2$ to $\mathbb{S}^2$

I am currently working on research involving packing problems and am finding myself needing the tools from Combinatorial Geometry (in particular, I've been reading Pach and Agarwal's book on the ...
7
votes
1answer
746 views

Derived functors of symmetric powers

What do the derived functors of the symmetric powers look like? I understand that this is related to the homology of the symmetric groups, but I don't know a reference for that. Namely, I'm ...
4
votes
1answer
283 views

Generalizing the circle packing theorem to 3-dimensions

The circle packing theorem (Koebe–Andreev–Thurston theorem) states that every finite planar graph is the nerve of some disk packing in the plane, where the nerve of a packing $P$ is a graph $G=(V,E)$, ...
7
votes
1answer
271 views

What is the homotopy type of a free simplicial ring?

Is there a good description of the homotopy type of a free simplicial ring (or simplicial $R$-algebra) on a given simplicial set, in terms of the homotopy type of that simplicial set? (This is mostly ...
8
votes
2answers
636 views

Connective spectra versus simplicial abelian groups - very basic question

Hello, I have very general , "introductory" questions (It is quite hard for me to seek for specific things in the algebraic topology literature). I guess that connective spectra have a model ...
5
votes
2answers
599 views

Reference request - CDGA vs. sAlg in char. 0

Hello, Are the model categories of simplicial commutative algebras over $k$ and that of commutative differential graded algebras (in negative cohmological dimension) Quillen equivalent in char. 0 (or ...
3
votes
1answer
152 views

Coskeleta of simplicial principal bundles

Let $X_{\bullet}$ be a simplicial topological space. There is a truncation functor $tr^n \colon Fun(\Delta^{op}, Top) \to Fun(\Delta_n^{op},Top)$ (where $\Delta_n$ is the full subcategory of $\Delta$ ...
2
votes
1answer
177 views

Simplicial complex made of central idempotents of an algebra

Let $A$ be an algebra, say over $\mathbb{C}$ and finite-dimensional, but not necessary semisimple. I have the strong feeling, which I would like to prove and use, about the following rather natural ...