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 $\bigcup_{K\in\mathcal{P}} K=X$, ...


24

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{...


23

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 ...


18

It's called a groupoid. Given an object $A$, call the degenerate edge from $A$ to itself the identity map at $A$. Given an edge $f:A\to B$, let $f^{-1}:B\to A$ denote the unique edge that fills in a 2-simplex whose other two edges are $f$ and the identity. Given $f:A\to B$ and $g:B\to C$, define $gf$ to be the unique edge that fills in a 2-simplex whose ...


17

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 ...


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

Iā€™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

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 ...


16

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/


16

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 $\...


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


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

One key point is that if you let $S/2$ denote the cofibre of twice the identity on the sphere spectrum (also known as the mod $2$ Moore spectrum, or $\Sigma^{-1}\mathbb{R}P^2$), then twice the identity map on $S/2$ is nonzero. This cannot happen in the homotopy category of simplicial abelian groups, or (roughly speaking) any other triangulated category ...


14

Marco Grandis has done some work on this, and you can extract answers for 1-3 from these papers Finite Sets and Symmetric Simplicial Sets - M Grandis - TAC (pdf) Higher Fundamental Functors for Simplicial Sets - M Grandis - CTGDC (pdf) See also An Alternative Presentation of the Symmetric-Simplicial Category - Eric R. Antokoletz - arxiv (link) Question 1 ...


14

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) ...


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

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

One fundamental difference concerns the behavior of Postnikov towers, or the relationship between the spectrum/simplicial abelian group and its homotopy groups. In simplicial abelian groups all Postnikov towers are splittable, since there are no higher Ext's between abelian groups; thus every simplicial abelian group is equivalent to a product of K(A,n)'s. ...


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

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. ...


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

A different proof, which works in much more generality, appears as part of Theorem 5.22 in my paper All (āˆž,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

I don't think there will be a simple necessary and sufficient condition. For exotic triangulations, the links $L$ of vertices will not be manifolds. One thing you need (not sufficient) is for $L \times \mathbb{R}$ to be a manifold. It's kind of shocking that this is possible for non-manifolds; the first example of it is Bing's dogbone space. A weaker ...


12

Let $X$ be any finite complex such that any map $X\to X$ homotopic to the identity is surjective and for which there is a surjective but nullhomotopic PL map $f:X\to X$ (for instance, $X$ could be $S^n$ for any $n>0$). Let $K$ be the mapping telescope obtained by iterating $f$. Then $K$ is contractible, locally compact, and finite-dimensional, but for ...


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

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

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 ...


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