For questions about simplicial sets (functors from simplex category to sets), simplicial (co)algebras and simplicial objects in other categories; geometric realization, model structures, Dold-Kahn correspondence etc. Please do not use for questions about geometry of simplices nor about ...

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18
votes
1answer
3k views

Difference between simplicial and singular homology?

I am having some difficulties understanding the difference between simplicial and singular homology. I am aware of the fact that they are isomorphic, i.e. the homology groups are in fact the same (and ...
11
votes
3answers
604 views

What is combinatorial homotopy theory?

Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below. Many times I've heard people ...
10
votes
1answer
211 views

How to construct a quasi-category from a category with weak equivalences?

Let $(\mathcal{C},W)$ be a pair with $\mathcal{C}$ a category and $W$ a wide (containing all objects) subcategory. Such a pair represents an $(\infty,1)$-category. One model for such gadgets is a ...
8
votes
2answers
532 views

What is the homotopy colimit of the Cech nerve as a bi-simplical set?

Let $f:Y_\bullet\to X_\bullet$ be an epimorphism of simplicial sets and define the bi-simplicial set $$ F_{\bullet\bullet}=\ldots Y\times_X Y\times_X Y\underset{\to}{\underset{\to}{\to}}Y\times_X Y ...
8
votes
1answer
308 views

How does hocolim relate to Hom?

In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like ...
8
votes
1answer
404 views

Toric Varieties: gluing of affine varieties (blow-up example)

Let $\Delta$ be a fan, consisting of cones $\sigma_0=conv(e_1,e_1+e_2)$ and $\sigma_2=conv(e_1+e_2,e_2)$ and $\tau=\sigma_0\cap\sigma_1=conv(e_1+e_2)$. The dual cones are $\sigma_0^\vee= ...
8
votes
1answer
60 views

When are the geometric realisations of two simplicial sets homeomorphic?

I'm given two simplicial sets $X,Y : \Delta^{\operatorname{op}} \to Set$. Of, course, if I study their geometric realisations $\lvert X \rvert$ and $\lvert Y \rvert$, I might find a homeomorphism, or ...
8
votes
1answer
122 views

Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic)

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
8
votes
0answers
144 views

Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
7
votes
1answer
249 views

Homology other than singular?

Usually, one defines $n$-th homology functor on topological spaces as the composite functor $$ \mathbf{Top} \to [\Delta^\mathrm{op},\mathbf{Set}] \to [\Delta^\mathrm{op},R\!-\!\mathbf{Mod}] \overset C ...
7
votes
1answer
94 views

Boundary of boundary of singular cube is zero (Spivak)

At the bottom of page 99 of M. Spivak's Calculus on Manifolds he arrives at the formula $$\partial (\partial c)=\sum_{i=1}^n \sum_{\alpha=0,1} \sum_{j=1}^{n-1} \sum_{\beta=0,1} ...
7
votes
0answers
110 views

History of the term “anodyne” in homotopy theory

There is a notion of an anodyne morphism (usually of simplicial sets). This type of morphisms is used, for instance, to establish basic properties of quasicategories. But the name is quite mysterious. ...
7
votes
0answers
117 views

A theorem of Kan regarding fibrant replacement

Recall that there is an adjunction $$\mathrm{Sd} \dashv \mathrm{Ex} : \mathbf{sSet} \to \mathbf{sSet}$$ where $\mathrm{Sd} (\Delta^n)$ is the first barycentric subdivision of $\Delta^n$. There is a ...
6
votes
2answers
205 views

How to intrinsically think about simplicial objects.

It seems that a simplicial set should be thought as a space/thing, with encoded building information. The set of $n$-simplicies is the set of n-dimensional pieces and the boundary and degeneracy maps ...
6
votes
3answers
263 views

Why is the 'mapping space' between two objects in a quasi-category a Kan complex?

Let $X$ be a quasi-category. (i.e. a simplicial set satisfying the weak Kan extension condition). Given two 'objects' $a,b\in X_0$ in $X$, one defines the mapping space $X(a,b)$ to be the pullback of ...
6
votes
1answer
116 views

cofaces and codegeneracies on Simplicial Sets

Let Δ be the category of finite ordinal numbers with order-preserving maps, i.e., Δ consists of objects strings A morphism f:n→[m] is an order-preserving function (a functor) and we can think of the ...
6
votes
1answer
233 views

Fat geometric realization weakly equivalent to the usual one

Let $X$ be a simplicial set. Recall that we can associate to it a certain topological space, the geometric realization, given by $ | X | = \int ^{n \in \Delta} X_{n} \times \Delta _{n} \simeq ...
6
votes
1answer
169 views

Does the category of Simplicial Complexes have finite limits and colimits?

Does the category of Simplicial Complexes have finite limits and colimits? Does the geometric realization functor preserve them? Thanks!
6
votes
1answer
174 views

Cohomological Whitehead theorem

Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...
6
votes
1answer
165 views

Geometric realization of function complexes of simplicial sets

Let $\operatorname{sSet}$ be the category of simplicial sets and $\operatorname{Top}$ the category of (say, compactly generated) topological spaces. There is a pair of adjoint functors called ...
6
votes
1answer
179 views

Inverse functor in proof of Dold Kan Correspondence

I´m looking at the proof of the Dold-Kan correspondence. Let $SA$ be the category of simplicial objects in an abelian category $A$ and $CH_{\ge0}(A) $ the category of non-negative chain complexes in ...
6
votes
0answers
174 views

Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$

I was wondering if I am in the right track here. Let $A := k[x,y]/(x^2 - y^3)$, the cuspidal curve. Obviously this isn't etale or smooth over $k$ so its cotangent complex is not contractible. Now, I ...
6
votes
0answers
137 views

What is a copresheaf on a “precategory”?

Let $\mathcal{S}$ be a category with pullbacks. A precategory $\mathbb{C}$ in the sense of Borceux and Janelidze [Galois Theories, §7.2] comprises three objects $C_0, C_1, C_2$ in $\mathcal{S}$ and ...
5
votes
2answers
136 views

Are bounded open regions in $\mathbb{R}^n$ determined by their boundary?

Let $U$ and $V$ be two bounded open regions in $\mathbb{R}^n$, and let us further assume that their topological boundaries are nice enough that they are homeomorphic to finite simplicial complexes. ...
5
votes
2answers
318 views

Passing pullbacks through adjunction

I'm having trouble following the proof of Proposition I.5.2 in Goerss-Jardine (Simplicial Homotopy Theory). After establishing the adjunction $\hat\Delta(X\times K,Y) \simeq \hat\Delta(K,[X,Y])$, ...
5
votes
1answer
212 views

What does it mean for a category to be “tensored over” another category?

What does it mean for a category to be "tensored over" another category? I was reading "Stable model categories are categories of modules" by Schwede and Shipley ...
5
votes
2answers
267 views

Excision via simplicial sets

Let $X$ be a topological space, covered by a collection of open sets $\{U_\alpha\}$ (or more generally a cover by subspaces whose interiors cover $X$). Consider the singular simplicial set ...
5
votes
1answer
1k views

Barycentric subdivisions of simplices yield a simplicial complex

The following interesting result (in particular parts (b) and (d)) is stated either as a obvious fact or as an exercise in several books on algebraic topology: The barycenter $b_\sigma$ of an ...
5
votes
2answers
140 views

Model structure on sSet

Which is the model structure on $ \text{sSet} $ (category of simplicial sets) that makes $\text{sSet}$ Quillen equivalent to the category $ \text{Cat} $ (of small categories) by the adjunction ...
5
votes
1answer
314 views

What are simplicial topological spaces intuitively?

(NOTE: I reposted the question to MO. Please answer there.) I tend to imagine simplicial objects in a category as some kind of "topological objects", with a notion of homotopy. Simplicial sets are ...
5
votes
1answer
231 views

The two-sided simplicial bar construction is Reedy-cofibrant

Let $\mathcal{M}$ be a simplicial model category and let $\mathbb{C}$ be a small category. For a diagram $F : \mathbb{C} \to \mathcal{M}$ and a weight $G : \mathbb{C}^\mathrm{op} \to \mathbf{sSet}$, ...
5
votes
1answer
267 views

What exactly is the CW complex structure on a geometric realisation?

This is likely a silly question. Definitions: $\bullet$ $\Delta_n = \{ (t_0, \dots, t_n) \: | \: 0 \leq t_i \leq 1, \sum_i t_i = 1 \}$ $\bullet$ Given $f: \underline{m} \to \underline{n}$ in ...
5
votes
1answer
186 views

Pseudomanifolds

An $n$-dimensional (closed) pseudomanifold is a finite simplicial complex $X$ such that (i) every simplex is a face of an $n$-simplex (ii) every $(n-1)$-simplex is a face of exactly two ...
5
votes
0answers
36 views

Replacing faces of a cube in a quasicategory

I have a question on quasicategories which seems to be unavoidably cubical, and which I haven't been able to locate any information about. Suppose I have a cube $U:\mathbf{2}^3:\to C$ in a ...
5
votes
0answers
27 views

The dimension of $H_i(X_n, \mathbb{Q})$ is linear in $n$ with a bounded nonnegative error, where the error is periodic?

Let $X$ be a finite simplicial complex with a simplicial map to $S^1$. Take covers of $X$ associated to the subgroups $n \mathbb{Z}$ of $\mathbb{Z} = \pi_1(S^1)$. This defines a sequence of spaces ...
5
votes
1answer
69 views

Cech Cohomology and the Dold-Kan Correspondence

Given a (co/contravariant) functor $F$ from the simplicial category $\Delta$ to an abelian category $A$, we can form its Cech complex (or "alternating face map complex" on the nLab), i.e. ...
5
votes
0answers
122 views

Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the simplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
5
votes
0answers
178 views

Two definitions of equivariant sheaves

Let $G$ be a topological group. Here are two definitions of $G$-equivariant sheaves on a $G$-space $X$. (a) Define an $G$-equivariant sheaf by a sheaf $F$ (étalé space) equipped with a $G$-action ...
4
votes
1answer
280 views

Why do we ask that condition in Kan complex?

Let $\{X_n\}_{n=0}^\infty$ be simplicial set with faces $d_i:X_n\to X_{n-1} $, a simplicial set is called a Kan Complex if for any $x_0,...,x_{k-1},x_{k+1},...,x_{n+1}$ if $d_i(x_j)=d_{j-1}(x_i)$ for ...
4
votes
2answers
348 views

What's stopping me from choosing the nth Eilenberg Mac Lane space to be the following simplicial abelian group?

Given an abelian group $X$, let $F_n(X)$ denote the simplicial abelian group defined as follows: $F_n(X)_j=0$ for all $j<n$ and $F_n(X)_j=X$ for all $j\geq n$ with the appropriate zero and ...
4
votes
2answers
228 views

'homotopy' between morphisms of a 'topological' or 'algebraic' category (Stanley-Reisner ring)

In what follows, a homotopy is a congruence $\simeq$ on a given category. Given such a homotopy, objects $X$ and $Y$ of the given category are homotopy equivalent when there exist morphisms ...
4
votes
2answers
421 views

Kan fibrations and surjectivity

I have a basic question on the usual model structure on simplicial sets. What is the relation between being a Kan (trivial maybe ?) fibration and surjectivity ? Surjectivity here means either ...
4
votes
2answers
233 views

How do we get a simplicial homology functor?

The $n$-th simplicial homology group $H_n(A)$ of an abstract simplicial complex $A$ depends on the choice of an orientation for $A$ (but for different orientations, the homology groups are ...
4
votes
2answers
205 views

Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ preserve (homotopy)colimits?

Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ given by $S(X)_m=Hom_{\mathbf{Top}}(\Delta^m,X)$ preserve colimits? If not, what is a counterexample? The only things I can say are that it ...
4
votes
2answers
223 views

The empty set in homotopy theoretic terms (as a simplicial set/top. space)

I am currently confused about the empty set in terms of its path components and how this fits into the Quillen adjunction between topological spaces and simplicial sets. Probably, one of my ...
4
votes
1answer
38 views

Question about limit of cosimplicial diagram associated with a sheaf

Let $F$ be a presheaf of sets on the usual topological site. Let $\left\{U_i\rightarrow U\right\}$ be covering of $U$. On the second page of Hypercovers and Simplicial Presheaves the authors write the ...
4
votes
1answer
823 views

Does the homology, homotopy, and geometric realization functors of a simplicial group preserve colimits?

Given a based simplicial group, you can find its reduced homology with coefficients in a field, homotopy, and geometric realization. These are functors. If I have a free product of based simplicial ...
4
votes
1answer
186 views

Cech cohomology and cohomology of a category : a cluster of questions.

I apologize in advance : what follows is a bit of a mess. Also, I think it might be a big tautology, but i don't see it yet. My question is about the rapport of Cech cohomology and cohomology of a ...
4
votes
1answer
107 views

Existence and homotopies of embeddings between simplicial complexes

Let $K$ and $L$ be simplicial complexes, $m=\dim K$, and $h:|K|\rightarrow |L|$ be a continuous map. Show that $h$ is homotopic to a map carrying $K$ into $L^{(m)}$, the $m$-skeleton of $L$. I'm ...
4
votes
1answer
226 views

The Fundamental Group of a Polyhedron Depends Only on its $2$-Skeleton

Does anyone have a good quick proof of this using the Simplicial Approximation Theorem? I'm aware that it comes out as a corollary when considering edge paths and the edge group, but this seems like ...