30

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 $X$ is a simplicial set, and $\mathcal{P}$ a finite collection of subobjects of $X$ which is closed under intersections, and the $\bigcup_{K\in\mathcal{P}} K=X$, ...


25

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 $s\mathrm{Ab}=\mathrm{Fun}^{\mathrm{add}}(\mathbb{Z}\Delta^\mathrm{op}, \mathrm{Ab})$, the category of additive functors. Let $\mathcal{C}$ be the "Karoubi envelope" of $\mathbb{...


24

If $\mathcal{A}$ is an abelian category, then the Dold-Kan correspondence supplies an equivalence between the category of simplicial objects of $\mathcal{A}$ and the category of nonnegatively graded chain complexes in $\mathcal{A}$. One can therefore think of simplicial objects as a generalization of chain complexes to non-abelian settings. In homological ...


19

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 $\theta^n_{u,t}:\Delta_n\to\Delta_n$ by $\theta^n_{u,t}(x)=tx+(1-t)u$. Then define $\phi_t[\sigma,u]=[\sigma\circ\theta^n_{u,t},u]$. To see that this is well-defined, suppose that $\...


19

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 of the Seifert-van Kampen theorem. More precisely, Proposition A.3.2 in Higher Algebra says that that the "underlying homotopy type" functor $$Sing: Open(X) \to ...


18

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 to unpointed spaces is a right Quillen adjoint. That is, the homotopy limit of a diagram of pointed spaces is the same whether taken in the pointed or unpointed ...


17

Let $\mathscr C$ be a small category. Necessary and sufficient conditions for a presheaf $F$ to be cofibrant in the global projective model structure on $[\mathscr C^\mathrm{op}, [\Delta^\mathrm{op}, \mathbf{Set}]]$ are that: (1) Each $F(-)(n) \colon \mathscr C^\mathrm{op} \to \mathbf{Set}$ is projective (i.e., a coproduct of retracts of representables; if ...


17

The simplicial set itself does not give much (most varieties don't have very many maps from affine spaces), but Suslin introduced something along these lines, using maps from algebraic simplices to symmetric powers of $X$ as algebraic-geometry versions of singular chains. Here is a paper by Suslin and Voevodsky: http://www.math.uiuc.edu/K-theory/0032/


17

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 familiar with CW-complexes and several of the major theorems about them already which will be generalized (e.g. the Whitehead theorem). The reader is assumed to be ...


17

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 $$ H_n(X;\mathbb{Z}) \cong [ S^{n+t} , X\wedge K(\mathbb{Z}, t)] $$ for $t$ large. Then the Hurewicz map is the map induced by the generator $g:S^t \to K(\mathbb{Z}, t)$: $$ \...


16

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 by a functor $F : C^{op} \to Top$? One solution to this (which I learnt from Section 7 of S. Galatius, I. Madsen, U. Tillmann, M. Weiss, "The homotopy type of ...


16

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, $BG$, which is the presheaf of ∞-groupoids on $C$ presented by $B_\bullet G$. The precise relation between these three objects is the following: $\mathbf BG$...


15

As you point out (relayed from Frank Lutz), it seems likely that checking shellability is NP-hard. But all is not lost: A complex that is shellable usually has lots of shellings, and it's often quick to find them by recursively trying to extend a partial shelling. The above-mentioned answer mentions some ways that this can be made more efficient. A (pure) ...


15

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 morphism between any pair of objects. The nerve functor sends equivalences of groupoids to homotopy equivalences of simplicial sets, and the geometric ...


14

First question$\newcommand{\op}[1]{{#1}^{\mathrm{op}}}$$\newcommand{\sSet}{\mathrm{sSet}}$$\newcommand{\Grpd}{\mathrm{Grpd}}$$\newcommand{\Cat}{\mathrm{Cat}}$$\newcommand{\NN}{\mathbb{N}}$$\newcommand{\sres}{\mathrm{sres}}$$\newcommand{\hocolim}{\operatorname{hocolim}}$$\newcommand{\diag}{\operatorname{diag}}$$\newcommand{\To}{\longrightarrow}$$\newcommand{\...


14

The term geometric realization is used in HTT to refer to colimits indexed by $\Delta^{op}$. So an object $P \in \mathcal{C}$ is projective if and only if, for every simplicial object $X_{\ast}$ in $\mathcal{C}$, the canonical map $$ \varinjlim \text{Map}(P, X_{\ast} ) \rightarrow \text{Map}(P, \varinjlim X_{\ast} )$$ is a homotopy equivalence of spaces. ...


14

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. The Adem relations follow from an extra symmetry (transpose) of the double composition $X \to X^4$.


13

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 $n$ vertices is at least $$\exp \left( \exp \left( 0.004n \right) \right),$$ for all large enough $n$. This matches the upper bound from Dedekind numbers, up to the ...


13

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 $X_1 = \{0, 1, 1/2, 1/3, \ldots\} \subset \Bbb R$, with degeneracy $s^0: X_0 \to X_1$ being the natural inclusion. Build the rest of $X$ so that all higher simplices ...


13

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 adjunction $sk_k \dashv cosk_k : sSet \to sSet$. $cosk_k$ preserves the property of being a quasicategory, i.e. descends to a functor $cosk_k : qCat \to qCat$. To ...


13

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 cofibration $X \to Z$ followed by a fibration $Z\to B$, deduce from right properness and 2-out-of-3 that $X\to i^*Z$ is a weak equivalence, then use the ...


12

Since there were no answers for a few months, I asked this question to my colleague and triangulation expert Frank Lutz. Since his response was wonderful and exhaustive, I am reproducing it here for the benefit of others who find such matters interesting. Spoiler alert: it is very hard to test for shellability. Testing shellability is a mess. The ...


12

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


12

The answer is sort of well known. A simplicial set $X$ is the nerve of a groupoid if and only if any $n$-horn has a unique filler for all $n\geq 2$. The horn $\Lambda^k[n]$ is the simplicial subset of $\Delta[n]$ obtained by removing the non-degenerate $n$-simplex of $\Delta[n]$ and it's $k^{\text{th}}$ face. The previous statement means that any map $\...


12

In the topological category the usual compact-open topology does the job. (EDIT: Or rather I suppose you might have to modify it a little so that $h\mapsto h^{-1}$ is continuous.) At times you might want to replace a function space by the simplicial set called its total singular complex. This is an instance of the more general procedure of replacing a space ...


12

Here is the $1$-type case. I assume all spaces are of the homotopy type of CW. Let me write $haut(X)$ (resp. $haut_*(X)$) for the monoid of self-equivalences (resp. pointed ones) to avoid posible confusion with the group-theoretic notation. These spaces have the correct homotopy type by our assumption. From All Groups are Outer Automorphism Groups of Simple ...


12

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 recently wrote a paper where we solved a problem, which is closely related to your question. Namely, we give an algorithm which for a given finite simply connected ...


12

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 formally inverting the markings in question. In more detail: The forgetful functor $\mathrm{Set}^+_{\Delta} \to \mathrm{Set}_{\Delta}$ is a (right) Quillen ...


12

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


12

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 $$a_{A,B}^* \colon \mathbf{Cyl}(A,B) \longrightarrow \mathbf{sSet}/(A^\mathrm{op} \times B)$$ which is both a left and a right Quillen equivalence between the Joyal model structure on $\mathbf{...


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