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31 votes
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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 ...
Tim Campion's user avatar
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30 votes
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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 - ...
Simon Henry's user avatar
  • 40.5k
19 votes
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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 ...
Gregory Arone's user avatar
19 votes
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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 ...
Peter LeFanu Lumsdaine's user avatar
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})...
Jeff Strom's user avatar
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17 votes
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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 ...
Dmitri Pavlov's user avatar
17 votes
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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 ...
Maxime Ramzi's user avatar
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16 votes
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$BG$ 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,...
Marc Hoyois's user avatar
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16 votes
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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 ...
Denis Nardin's user avatar
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15 votes
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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 ...
Reid Barton's user avatar
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.
Stephan Zhechev's user avatar
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. ...
Jeff Strom's user avatar
  • 12.5k
14 votes
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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 ...
Tim Campion's user avatar
  • 62.2k
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 ...
Simon Henry's user avatar
  • 40.5k
13 votes
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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 $n$ ...
Matthew Kahle's user avatar
13 votes
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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 ...
Stephan Zhechev's user avatar
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 ...
Mike Shulman's user avatar
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13 votes
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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 ...
David White - gone from MO's user avatar
12 votes
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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 ...
Dylan Wilson's user avatar
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12 votes
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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" ...
Alexander Campbell's user avatar
12 votes
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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 @...
Tim Campion's user avatar
  • 62.2k
12 votes
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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,...
Alexander Campbell's user avatar
11 votes
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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 ...
Yonatan Harpaz's user avatar
11 votes
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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 ...
Charles Rezk's user avatar
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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 ...
Tom Mrowka's user avatar
  • 3,014
11 votes
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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 ...
S. carmeli's user avatar
  • 4,074
11 votes
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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 ...
D.-C. Cisinski's user avatar
11 votes
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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.
Todd Trimble's user avatar
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11 votes
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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 ...
Connor Malin's user avatar
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