30
votes

Accepted

### Errata on Rezk's paper

It looks like I completely missed this.
Here's what I guess happens: although the original 2.19 was wrong, there is a weaker version that is true (I'll just state it for simplicial sets): If $X$ ...

Community wiki

28
votes

Accepted

### Why are quasi-categories better than simplicial categories?

As a preface, I think that this question should be viewed as analogous to "what are the advantages of ZFC over type theory" or vice versa. We're talking about foundations -- in principle, it ...

25
votes

### A more natural proof of Dold-Kan?

The proof of the Dold-Kan theorem basically amounts to the following. Let $\mathbb{Z}\Delta$ denote the pre-additive category generated by the simplicial indexing category, so that $s\mathrm{Ab}=\...

19
votes

Accepted

### Why does the singular simplicial space geometrically realize to the original space?

You can just write down the required homotopy.
A point in $|\text{Sing}(X)|$ is an equivalence class $[\sigma,u]$ where $u\in\Delta_n$ and $\sigma:\Delta_n\to X$. Define $\theta^n_{u,t}:\Delta_n\...

19
votes

Accepted

### When is a topological space the homotopy colimit of an open covering?

It is true in complete generality that $X$ is the homotopy colimit of $C_U$ (and hence that the fat realization computes the homotopy colimit in this case). This is a special case of Lurie's version ...

18
votes

Accepted

### About definition of homotopy colimit of Kan and Bousfield

You can define homotopy limits and colimits in pointed as well as unpointed spaces. It so happens that the two notions of homotopy limit coincide, basically because the forgetful functor from pointed ...

17
votes

Accepted

### P.G.Goerss, J.F.Jardine, "Simplicial Homotopy Theory" prerequisites

As the commenters already argued, I would not regard this book as a self-contained introduction. For instance, from a brief browse through the introductory chapters:
The reader is assumed to be ...

17
votes

### An abstract nonsense proof of the Hurewicz theorem

I’d argue that it boils down to the generator $S^n\to K(\mathbb{Z},n)$ being an $(n+1)$-equivalence.
More detail: If you take the represented version of homology, it is given by
$$
H_n(X;\mathbb{Z})...

17
votes

Accepted

### Classifying space BG and contractable space EG

The easiest way to construct an explicit contracting homotopy
is to observe that EG is the geometric realization of the nerve of the groupoid G//G,
which has G as its set of objects and exactly one ...

16
votes

Accepted

### Topological Grothendieck Construction

This is a standard irritation. The issue is that $Top$ is not a category internal to $Top$, because it doesn't have a space of objects (and I don't mean for set-theoretic reasons), so what do you mean ...

16
votes

Accepted

### $BG$ the stack, $BG$ the simplicial presheaf

The two constructions are not quite equivalent. Let me write $\mathbf BG$ for the stack and $B_\bullet G$ for the simplicial scheme to better distinguish between them. There is a third relevant player,...

15
votes

Accepted

### Algebraic topology and homotopy theory with simplicial sets instead of topological spaces

It depends on what you mean by "all results". Of course results regarding manifolds or vector bundles do not admit statements completely internal to the world of simplicial sets (although ...

14
votes

### Teaching Steenrod Operations

I like to observe that the diagonal map $X\to X\times X$
is $\mathbb{Z}/2$-equivariant, hence induces a map of homotopy colimits.
Analyzing this map with $X = K( \mathbb{Z}/2,n)$ gives the squares.
...

13
votes

Accepted

### Is the geometric realization of a level-wise weak equivalence a weak equivalence?

To get such a result we typically need that the degenerate subspaces include via cofibrations, and we can get a counterexample by picking a standard non-cofibration.
Let $X_0 = \{0\}$, and let $X_1 = ...

13
votes

Accepted

### How many simplicial complexes on n vertices up to homotopy equivalence?

Andrew Newman just posted a preprint to the arXiv showing that the answer is doubly exponential in $n$.
In particular, he showed that the number of homotopy types of simplicial complexes on $n$ ...

13
votes

### Testing simplicial complexes for shellability

In a very recent work (https://arxiv.org/abs/1711.08436) it was shown that deciding shellability is NP-complete.

13
votes

Accepted

### What is the coskeleton tower of a quasi-category?

It turns out the answer is yes: $k$-coskeletalization of a quasicategory models truncation of an $(\infty,1)$-category to a $(k-1,1)$-category.
Let's collect some easy observations.
We have an ...

13
votes

### Why is Kan's $Ex^\infty$ functor useful?

Another thing for which Kan Ex$^{\infty}$ functor is useful is actually in the construction of the Kan-Quillen model structure. It morally gives a purely algebraic version of the simplicial ...

13
votes

### Extending Kan fibrations, without using minimal fibrations

A different proof, which works in much more generality, appears as part of Theorem 5.22 in my paper All (∞,1)-toposes have strict univalent universes: factor the composite $X\to A\to B$ as an acyclic ...

12
votes

Accepted

### Realisation of maps between spheres by simplicial maps

First, $\pi_{4k-1}(S^{2k})$ has an infinite cyclic direct summand for every $k \geq 1$. As a simple example, you can think of $\pi_3(S^2) \cong \mathbb{Z}$, coming from the Hopf fibration.
We ...

12
votes

Accepted

### On HTT's Lemma 3.3.4.1

The lemma is true as stated. The Cartesian equivalence in question does not imply an equivalence in the Joyal model structure after forgetting the markings; rather, it implies an equivalence after ...

12
votes

Accepted

### Higher Topos Theory Theorem 2.2.5.3

From naturality we have the following commutative diagram:
$\require{AMScd}
\begin{CD}
Map_{\mathfrak C[S]}(x,y) @<\sim<< |Hom^R_S(x,y)|_{Q_\bullet} @>\sim>> Hom^R_S(x,y) \\
@VVV @...

12
votes

Accepted

### Correspondences of $\infty$-categories

Your question is answered by the following result, for which I will give a few references.
Theorem. For each pair of simplicial sets $A$ and $B$, there is a functor $$a_{A,B}^* \colon \mathbf{Cyl}(A,...

11
votes

Accepted

### What is this symmetric simplex category, concretely?

$\Delta_+$ is the monoidal category generated from the associative operad, considered as a non-symmetric operad. Similarly, $(\Delta_+)_{{\rm sym}}$ is the symmetric monoidal category generated from ...

11
votes

### Teaching Steenrod Operations

Its nice to look also at Bott's early paper
"On symmetric products and the Steenrod squares. "
Ann. of Math. (2) 57, (1953). 579–590.
He uses an early version of Smith theory. Depending on how
you ...

11
votes

Accepted

### Homotopy type of the semi-simplicial set of symmetric groups

It is contractible. To see this first observe that it is simply connected. It has 2 arcs $a = (1,2)$ and $b = (2,1)$, however the tirangle $(3,1,2)$ gives us the relation that $ab = b$ and ...

11
votes

Accepted

### Weak complicial sets: Are the morphisms too strict?

Indeed there is no such coherence result: it is false already for $2$-categories (see for instance Lemma 2 of this paper of Steve Lack). The solution to your troubling corollary is that the "correct" ...

10
votes

Accepted

### Is the hom-simplicial set in the hammock localization a nerve?

The nLab description is not correct.
For each "shape" of zig-zag, there is a "hammock category" for it (not a groupoid, and the nLab page I am looking at never mentions groupoids here), whose ...

10
votes

Accepted

### Where to find the proof that these two version of simplicial homotopy are equivalent?

Proposition 6.2 in Chapter 1 of "Simplicial objects in algebraic topology", by J.P. May.

10
votes

Accepted

### Proper model category of simplicial rings revisited

This paper proves some things about left properness for categories of simplicial algebras. The context of the paper is in terms of "algebras for a simplicial algebraic theory", which certainly ...

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