13
votes
1answer
301 views

Higher Cerf Theory

Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that ...
2
votes
0answers
89 views

When is the product of an infinite family of simplicial sets also a homotopy product?

The homotopy product of an infinite family of simplicial sets can be computed by deriving the product functor sSetW→sSet, for example, by performing the componentwise fibrant replacement using Kan's ...
4
votes
0answers
146 views

Getting a Postnikov Tower from the Tot-tower?

I have had a problem I've been thinking of recently but can't seem to make anything of it. Let $X$ be a simplicial set. It is well known that one can construct a Postnikov tower $$P_nX \rightarrow ...
2
votes
0answers
94 views

Recognizing Simplicial (Quasi)Fibrations

Let's say we are given two finite simplicial complexes, which I will suggestively call $E$ and $B$. We'd like an algorithm for the following decision problem: Does there exist a simplicial map ...
4
votes
1answer
135 views

In a Simplicial group the Degenerate elements don't intersect the kernel of the face maps.

Let $G_{\bullet}$ be a simplical group. I denote by $D_n \subset G_n$ the subgroup generated by all the degeneracy maps $s_i:G_{n-1} \to G_n$. I also denote $$M_n = \bigcap_{i>0} \mathrm{Ker} d_i ...
3
votes
2answers
236 views

Pseudo-manifolds and homology

Is there a good reference for the proof that the cobordism group of pseudo-manifolds is isomorphic to the singular homology group? I was looking for a more geometrical definition of homology and ...
9
votes
1answer
428 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 ...
9
votes
2answers
386 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 ...
8
votes
3answers
479 views

References for the “nerve of an algebraic variety”

Let's do algebraic geometry over an arbitrary base ring $k$. I've frequently seen a definition of the algebraic $n$-simplex, as follows: $$\Delta^n = ...
2
votes
1answer
174 views

A certain kind of simplicial complex

I'm interested in collections $\mathcal{C}$ of tuples $\mathbf{t} = (n_1, n_2, \ldots, n_r)$ of positive integers satsifying if $\mathbf{t}\in \mathcal{C}$ then so is any permutation of $\mathbf{t}$ ...
6
votes
1answer
224 views

Are there models for homotopy colimits and limits of simplicial sets that generalize Kan's suspension and loop functors?

Consider the category C of pointed simplicial sets. The pair of functors X∈C↦X∧S¹∈C and Y∈C↦Map(S¹,Y)∈C models the suspension and loop functors on the underlying ∞-category of C. There is another ...
9
votes
0answers
139 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 ...
11
votes
0answers
410 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 ...
8
votes
1answer
308 views

Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid

Let $C$ be a category with an object $X$ such that there are no non-trivial endomorphisms $X\rightarrow X$. Consider a simplex $\sigma$ of the nerve $NC$ of $C$. It is just a string of composable ...
2
votes
1answer
338 views

A question about the proof of Quillen's Theorem A

(I posted this question on Mathstack but I haven't received any answers or comments so I thought I might as well try my luck here. I apologize if it is not an appropriate question.) Theorem (Quillen) ...
9
votes
1answer
244 views

Equivariant homotopy, simplicially

It is a classic result of Kan that the homotopy categories (with appropriate model structures) of simplicial sets and of topological spaces (in fact, one could only care about CW-complexes) are ...
9
votes
1answer
739 views

What are simplicial topological spaces intuitively?

(This is a repost of a question from MSE. I hope there is more to say.) I tend to imagine simplicial objects in a category as some kind of "topological objects", with a notion of homotopy. Simplicial ...
1
vote
0answers
92 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
106 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 ...
3
votes
0answers
103 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 ...
1
vote
2answers
284 views

Finite simplicial sets

A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. It is not hard to show that every finite simplicial set has only a finite number of simplicies in ...
9
votes
1answer
283 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 ...
12
votes
2answers
550 views

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 ...
7
votes
2answers
334 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 ...
-1
votes
1answer
82 views

Poset complex of reverse ordering [closed]

This might be too easy but I cannot proof it easily. Any reference or hint will be great. Q: Suppose P is a poset in which every chain is finite and $\Delta P$ is the poset complex associated to it. ...
3
votes
0answers
175 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 ...
3
votes
0answers
193 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 ...
1
vote
1answer
174 views

Finding automorphism groups of simplicial complexes

Question: Given a finite simplicial complex $K$, what general techniques allow one to efficiently compute (a presentation of) the group $\text{Aut}(K)$ of $K$'s automorphisms? Since this is ...
5
votes
0answers
186 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 ...
4
votes
1answer
231 views

Is there a class of simplicial sets whose weak homotopy type is preserved by symmetrization?

We have the categories, $S$, of simplicial sets and $SS$, of symmetric simplicial sets (whose simplices are unordered). There are functors: $H:S\to SS$ forgetting the ordering on simplices and ...
6
votes
1answer
391 views

Can we invert barycentric subdivision?

With apologies to fellow algebraic topologists, I confess that I have no idea how to answer this innocent-looking question: (1) Let's say we know that a finite simplicial complex $S$ is the ...
4
votes
0answers
169 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
2answers
324 views

Removing a simplicial subset from a simplicial set

Let $A, X$ be simplicial sets, and suppose there's an inclusion $A \longrightarrow X$. Geometrically realizing the inclusion map, we get a pair of spaces $(\mathcal{A}, \mathcal{X})$. I want to find ...
3
votes
0answers
205 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 ...
1
vote
1answer
166 views

How to call a simplicial set where every boundary of a simplex can be filled?

What is the correct terminology for the following property of a simplicial set $X_\bullet$: For every $k\geq 0$, every map $\partial\Delta^k\to X_\bullet$ can be extended to a map $\Delta^k\to ...
1
vote
1answer
115 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
225 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
562 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
230 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 ...
7
votes
3answers
449 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
386 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 ...
1
vote
1answer
202 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 ...
1
vote
0answers
165 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 ...
4
votes
1answer
334 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 ...
9
votes
2answers
422 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 ...
1
vote
1answer
257 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 ...
1
vote
1answer
93 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
55 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 ...
10
votes
0answers
277 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 ...