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 ...
- 56.6k
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 $X$ ...
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 - ...
- 38.5k
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 ...
- 18.4k
18
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 ...
- 50k
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 ...
- 10.3k
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})...
- 12.3k
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 ...
- 34.4k
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,...
- 8,432
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 ...
- 16.1k
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 ...
- 24.5k
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.
- 446
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.
...
- 12.3k
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 ...
- 56.6k
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$ ...
- 7,645
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 ...
- 446
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 ...
- 38.5k
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 ...
- 63.6k
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 ...
- 13.1k
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 @...
- 56.6k
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,...
- 5,205
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 ...
- 9,086
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 ...
- 2,904
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 ...
- 4,014
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" ...
- 5,205
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 ...
- 4,454
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 ...
- 26.4k
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.
- 13.1k
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 ...
- 26.4k
10
votes
Extending Kan fibrations, without using minimal fibrations
Yes, see Corollary 7.7 in Sattler's The Equivalence Extension Property and Model Structures.
- 7,526
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