Questions tagged [simplicial-stuff]
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-Kan correspondence etc. Please do not use for questions about geometry of simplices nor about triangulations.
545
questions
47
votes
1answer
8k 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 ...
17
votes
2answers
467 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 ...
14
votes
0answers
547 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 $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
13
votes
3answers
1k 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 speaking ...
11
votes
3answers
817 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
...
10
votes
1answer
3k views
Product of simplicial complexes?
Given two abstract simplicial complexes $\mathcal{K}$ and $\mathcal{L}$, what is the definition of their product $\mathcal{K} \times \mathcal{L}$, as another abstract simplicial complex? Basically I'm ...
10
votes
1answer
178 views
Higher categories for category theorists?
I'm not actually a category theorist, but assume I have some background in category theory and am pretty comfortable with it; and I want to learn higher category theory (say in the sense of Boardmann-...
10
votes
0answers
156 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 ...
9
votes
3answers
529 views
Right Kan extension of $\mathcal{F} : \mathsf{\Delta} \rightarrow \mathsf{Top}$.
My question arose while studying something about Kan Extensions.
We know that we have the following diagram
$$
\begin{array}{ccc} &&\mathsf{\Delta} & \xrightarrow{\mathcal{F}} & \...
9
votes
1answer
757 views
Fat geometric realization weakly equivalent to the usual one
Let $X$ be a simplicial set. Recall that we can associate to it a certain topological space, the geometric realization, given by
$ | X | = \int ^{n \in \Delta} X_{n} \times \Delta _{n} \simeq (\...
9
votes
1answer
808 views
Toric Varieties: gluing of affine varieties (blow-up example)
Let $\Delta$ be a fan, consisting of cones $\sigma_0=conv(e_1,e_1+e_2)$ and $\sigma_2=conv(e_1+e_2,e_2)$ and $\tau=\sigma_0\cap\sigma_1=conv(e_1+e_2)$. The dual cones are $\sigma_0^\vee= conv(e_2,e_1\!...
9
votes
1answer
242 views
Proof of Lemma 5.1.5.3 in Jacob Lurie's HTT.
I am currently trying to understand the following proof in Higher Topos Theory.
I am fine with almost all of the argument, except with the claim that $\mathcal{E}^1$ is a deformation retract of $\...
8
votes
2answers
876 views
What is the homotopy colimit of the Cech nerve as a bi-simplical set?
Let $f:Y_\bullet\to X_\bullet$ be an epimorphism of simplicial sets and define the bi-simplicial set
$$
F_{\bullet\bullet}=\ldots Y\times_X Y\times_X Y\underset{\to}{\underset{\to}{\to}}Y\times_X Y \...
8
votes
1answer
676 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!
8
votes
1answer
377 views
How does hocolim relate to Hom?
In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like this:$\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\colim}{colim}\DeclareMathOperator{\hocolim}{hocolim}\...
8
votes
1answer
192 views
A Mayer-Vietoris argument I don't get — From a paper by Bousfield-Gugenheim
I working through the article On PL De Rham theory and rational homotopy type by Bousfield-Gugenheim. I am having troubles completing the proof of lemma 5.3, where they claim that the fact that a ...
8
votes
1answer
222 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. ...
8
votes
1answer
98 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
155 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
285 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 ...
7
votes
3answers
1k 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 ...
7
votes
1answer
308 views
Boundary of a simplicial set in terms of a coequalizer
I am trying to understand why we have a coequalizer
$\sqcup_{0 \leq i < j \leq n} |\Delta^{n-2}| \rightrightarrows \sqcup_{0 \leq i \leq n} |\Delta^{n-1}| \rightarrow |\partial \Delta^n|$. What ...
7
votes
1answer
397 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
$$\partial (\partial c)=\sum_{i=1}^n \sum_{\alpha=0,1} \sum_{j=1}^{n-1} \sum_{\beta=0,1} (-1)^{i+\alpha+j+\beta}...
7
votes
1answer
553 views
Geometric realization of function complexes of simplicial sets
Let $\operatorname{sSet}$ be the category of simplicial sets and $\operatorname{Top}$ the category of (say, compactly generated) topological spaces. There is a pair of adjoint functors called ...
7
votes
1answer
467 views
Homology other than singular?
Usually, one defines $n$-th homology functor on topological spaces as the composite functor
$$ \mathbf{Top} \to [\Delta^\mathrm{op},\mathbf{Set}] \to [\Delta^\mathrm{op},R\!-\!\mathbf{Mod}] \overset C ...
7
votes
1answer
87 views
Homotopy (co)limit of sets
There are general constructions for homotopy (co)limit of diagrams in spaces
$$D:I \rightarrow Top$$
I was wondering if the homotopy (co)limit of discrete diagrams, i.e. when the spaces are all ...
7
votes
2answers
424 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 ...
7
votes
0answers
61 views
Is the nerve of a symmetric monoidal category a K-theory space?
It's an amazing fact that if the $C$ is a symmetric monoidal category so that its components form a group, then $NC$ is an infinite loop space. Now if we have a Waldhausen category $D$, a category ...
7
votes
0answers
156 views
Unique representation of a degenerate simplex
I recently learned about simplicial sets, and now I'm studying some basic properties. I've learned that every degenerate simplex is a degeneracy of a unique non-degenerate (nice) simplex. However, I ...
7
votes
0answers
126 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 point?...
7
votes
0answers
244 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 ...
6
votes
1answer
798 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 (http://www.math.uni-bonn.de/people/...
6
votes
1answer
917 views
Homology of a simplicial set
Let $X$ be a simplicial set. Define the complex $(C^X_\bullet,D)$ by $$C^X_n=\bigoplus_{X_n} \mathbb{Z}$$ and $$D_n=\sum_{i=0}^n (-1)^i d_i:C_n \to C_{n-1}$$ where the $d_i$'s are the face maps.
I ...
6
votes
2answers
296 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
1answer
849 views
Coboundary Formula
When $K$ is a simplicial complex, the dual complex $C^*(K)$ to the chain complex $C_*(K)$ has a concrete interpretation: an element in $C^n(K)$ (a cochain) is given by assigning an integer to every ...
6
votes
2answers
338 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 ...
6
votes
1answer
76 views
Issue with Coequalizer Definition of the Horn of a Simplex
I am having trouble understanding the maps in this coequalizer defining the k-horn (from page 9 of Goerss and Jardine's Simplicial Homotopy Theory). It is defining them using the ith, jth inclusion ...
6
votes
1answer
138 views
$\infty$-categories definition disambiguation
I am currently reading Lurie's - Higher Topos Theory, but am having difficulties from the very beginning. I know that the topic is too broad and different mathematicians may define differently some of ...
6
votes
1answer
209 views
(Co)homology of propositional logic
Sorry if this is a rather vague question, but it seemed like something that might be interesting.
Let $P$ be a family of propositions, and let $\mathcal L(P)$ be the set of all compound propositions ...
6
votes
1answer
493 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 ...
6
votes
1answer
505 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 $\...
6
votes
1answer
155 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. $CF^n=F([n])$...
6
votes
1answer
1k views
What is the cone over a simplicial set?
At the moment I'm reading through Edward B. Curtis, 'Simplicial Homotopy Theory' (Advances in Mathematics 6, 107-209 (1971)) in order to learn about simplicial sets and I run into a problem where the ...
6
votes
1answer
254 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 $...
5
votes
2answers
145 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
1answer
646 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 $...
5
votes
2answers
565 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
2answers
670 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 ...
5
votes
1answer
526 views
CW complex structure of geometric realization
In Ralph Cohen's notes on the topology of fiber bundles he makes the following claims:
on pp.69, he says the geometric realization of a simplicial set is a CW complex
on pp.70, he says the geometric ...
5
votes
1answer
297 views
Does the functor $\mathbf{cosk_n}:sSet\to sSet$ preserve Kan complexes?
On this nlab page a functor $\mathbf{cosk_n}:sSet\to sSet$ is constructed.
Is $\mathbf{cosk_n}(K)$ of a Kan complex $K$ again a Kan complex?
