31
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 ...

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

21
votes

Accepted

### Is there a higher analog of "category with all same side inverses is a groupoid"?

Yes, this is possible. The following is a classical result of the theory of quasi-categories (You'll find it in the early part of Lurie's Higher topos theory or in Joyal notes on quasi-categories - ...

19
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 ...

19
votes

Accepted

### What is known about the homotopy type of the classifier of subobjects of simplicial sets?

It’s not hard to check that the subobject classifier $\Omega$ is trivially fibrant, and so its homotopy type is trivial. Orthogonality of $\Omega \to 1$ against a map $i : A \to B$ corresponds to the ...

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 ...

17
votes

Accepted

### What are the centre and trace of the simplex category?

The center is trivial : as Benjamin said in the comments, an endomorphism of the identity is the identity on $\Delta^0$, and using the maps $\Delta^0\to \Delta^n$ you find that any endomorphism of the ...

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,...

16
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 ...

15
votes

Accepted

### Is the singular simplicial complex functor $\operatorname{Sing}_\bullet:\operatorname{Top} \to \operatorname{sSets}$ fully faithful for nice spaces?

Here is a simple counterexample with $X = Y = \mathbb{R}$:
Send a simplex $\sigma : |\Delta^n| \to \mathbb{R}$ to the affine function $F(\sigma) : |\Delta^n| \to \mathbb{R}$ with the same values at ...

14
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.

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.
...

14
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 ...

14
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

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

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 ...

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 ...

13
votes

Accepted

### Plus construction on Simplicial Sets?

The answer is yes. This is spelled out in the book The local structure of algebraic K-theory by Bjørn Ian Dundas, Thomas G. Goodwillie and Randy McCarthy. Check out Section 1.6.1 on page 26, where ...

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

### 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" ...

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

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 ...

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

### Modern proofs for simplicial localizations

For Question 1. It is documented in Corolary 4.2.4.8 in Lurie's Higher Topos theory that $N_\Delta(\underline{M}^{cf})$ is an $\infty$-category with small limits (and small colimits). Moreover, the ...

11
votes

Accepted

### Simplicial set construction of the classifying space

I believe that's called the Milgram bar construction:
R.J. Milgram, The bar construction and abelian $H$-spaces, Illinois J. Math. 11 (1967), 242-250.

11
votes

Accepted

### Homotopic but not equivariantly homotopic maps

For any $G$-space the $G$-equivariant maps $[EG,X]_G$, also known as the homotopy fixed points $X^{hG}$, are a Borel homotopy invariant of $X$ meaning that it is an invariant of $G$-equivariant maps ...

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