Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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.

795 questions
71
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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 maybe this doesnt't help my intuition), but I am…
user48168
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34
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Where does one learn how to apply categorical algebra and higher abstractions to algebraic topology?

Tl;Dr: I know higher category theory and algebra is used ubiquitously in advanced algebraic topology. However, every time I ask someone, or try to find out, how one actually learns to apply the higher theory to genuine topological problems (to have…
FShrike
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23
votes
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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 $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ following Bousfield and Kan; more precisely,…
Zhen Lin
  • 97,105
21
votes
2 answers

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 quasi-category (a simplicial set satisfying the weak…
KotelKanim
  • 2,986
14
votes
3 answers

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 about combinatorial homotopy theory, but every…
Giorgio Mossa
  • 18,698
13
votes
1 answer

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 (\bigsqcup X_{n} \times \Delta _{n}) / \sim$, where on…
Piotr Pstrągowski
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12
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1 answer

Product of simplicial complexes?

Given two abstract simplicial complexes $K$ and $L$, what is the definition of their product $K \times L$, as another abstract simplicial complex? Basically I'm looking for the definition of "product" such that $|K \times L|$ is homeomorphic to $|K|…
Vikram Saraph
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12
votes
1 answer

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-Vogt-Joyal-Lurie) but don't want to have to delve…
Maxime Ramzi
  • 45,086
12
votes
3 answers

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}} & \mathsf{Top}\\ &&\mathcal{y} \searrow& & \nearrow…
user321268
11
votes
3 answers

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 $$ X^{\Delta^1}\to X^{\partial\Delta^1}\cong…
KotelKanim
  • 2,986
11
votes
1 answer

Is the (non-normalized) Moore complex functor an adjoint? An equivalence?

Consider the Moore complex functor $M : \mathsf{sAb} \to \mathrm{Ch}^+(\mathsf{Ab})$, where the complex associated to a simplicial abelian group $A$ is $MA_n = A_n$ with boundary $\partial = \sum_{j=0}^n (-1)^j d_j$. If we look at the normalized…
Brendan Murphy
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10
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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 natural transformation $i :…
Zhen Lin
  • 97,105
10
votes
1 answer

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 $\sigma_0^\vee= conv(e_2,e_1\!-\!e_2)$ and $\sigma_1^\vee=…
Leo
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10
votes
1 answer

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. Why are such morphisms called anodyne? I checked…
Dmitry
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9
votes
2 answers

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 $\mathcal{E}$. While I feel like this is true, I am…
user187567
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