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17
votes
0answers
296 views

Is this a model for $K$-theory of a triangulated category?

The recent question Complete the following sequence: point, triangle, octahedron, . . . in a dg-category reminded me of something I wanted to clarify long time ago; most likely this is now well known ...
4
votes
1answer
514 views

Cech nerve as homotopy colimit?

Given a category $\mathcal{C}$ with a notion of covering $\{ U_{i} \rightarrow X \}$ for an object $X$ (say $\mathcal{C}$ is a Grothendieck site), we can form the Cech nerve $$ \cdots ...
3
votes
0answers
144 views

Complete Segal operads and dendroidal sets

There is a Quillen equivalence between the model category presenting Lurie's $\infty$-operads (which are inner fibrations $\mathcal{C}\to\mathrm{N}(\mathbf{F})$ satisfying certain conditions) and the ...
5
votes
1answer
172 views

What do globes (used to construct globular sets, $\omega$-categories, etc.) actually look like?

Nlab introduces the globular category as a geometrical model to construct certain higher categorical structures (e. g. strict $\omega$-categories), just as quasi-categories, for example, are modelled ...
5
votes
0answers
108 views

Two natural maps asssociated with the nerve of a cover

Let $X$ be a nice (e.g. paracompact, locally contractible) topological space, and let $\mathcal{U}=\{U_i\}_{i\in I}$ be an open cover of $X$. Also denote by $N$ the (topological realization of) the ...
12
votes
1answer
221 views

From relative categories to marked simplicial sets

Both relative categories and marked simplicial sets (over Δ^0) present the ∞-category of ∞-categories. Naturally, one could ask whether there is a reasonably direct way to pass between these two ...
11
votes
2answers
3k views

Understand Cech Cohomology

I am currently trying to understand Cech cohomology. Five questions arised and I would be glad for help. In what follows $X$ is a topological space. I really like Dugger's and Isaksen's paper ...
4
votes
0answers
113 views

Relative Hurewicz Theorem

For a given zero-reduced simply connected simplicial set $X$, one can define simplicial group $GX$ representing the loop space of $X$, its Abelianization $AX = GX/[GX,GX]$ and show that the map ...
1
vote
1answer
116 views

pullback square in Goerss-Jardine

In proving the the existence of the Reedy model structure on the category of simplicial objects in a model category $\mathcal{C}$, Goerss-Jardine prove there is a pullback square induced by a map of ...
-1
votes
1answer
106 views

Parallel transport on simplicial manifold? [closed]

Do you know some reference about the notion of parallel transport for simplicial manifolds?
4
votes
0answers
61 views

Homology of simplicial manifolds

Let $M_{\bullet}$ be a simplicial manifold. There are two ways to computing its cohomology. Consider the cosimplicial module $A_{DR}(M)$. It defines a functor $A_{DR}(M_{\bullet})\: : \: ...
5
votes
1answer
119 views

The properness of the special singular simplicial spaces

This is a question related to another one in MO Background : A special simplicial space $X_{\cdot}$ is a simplicial space with $X_{0}=\ast$ and $X_{n}\simeq X_{1}^{n}$ via the simplicial map ...
3
votes
1answer
92 views

Path space of a simplicial topological space?

Given a connected topological space $X$, its space of path $PX$ is again a topological space. On the other hand, for a simplicial set $K_{\bullet}$, its path space is given by $$ ...
6
votes
1answer
250 views

What's the best way to test if a sphere is a polytope? (algorithms for the Simplicial Steinitz Problem)

The problem of recognizing whether a simplicial face lattice is polytopal is sometimes called the Steinitz problem. Sturmfels and Bokowski advanced a set of methods in the late 80s to test whether ...
9
votes
1answer
218 views

Is the hom-simplicial set in the hammock localization a nerve?

Let $(C,w)$ be a relative category. Then associated to it we have its hammock localization, $L^H(C,w)$, which is a simplicially enriched category. If $X,Y\in C$, the description of the simplicial set ...
6
votes
0answers
163 views

Bott-Samelson theorem for simplicial sets

Let $X\in \mathrm{sSet}$ and $FX$ be the Milnor's construction (model for $\Omega\Sigma |X|$) - in each dimension $n$ this is the free group on $X_n$ with one relation $*=1$. I'm interested in ...
7
votes
1answer
353 views

What kind of category is a cyclically ordered set?

Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms ...
19
votes
2answers
647 views

How many simplicial complexes on n vertices up to homotopy equivalence?

Fix a number $n$, and define $\gamma(n)$ to be the number of simplicial complexes on $n$ unlabeled vertices up to homotopy equivalence. It is unlikely that an explicit formula exists, but what is ...
2
votes
1answer
352 views

Building $(\infty,2)$-categories from $\infty$-categories

Let $Y$ be a marked simplicial set, whose underlying simplicial set is also denoted by $Y$. Let $X$ be a scaled simplicial set such that the decalage of its underlying simplicial set is $Y$. $X$ is ...
15
votes
5answers
2k views

Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets?

Many topologists express a clear preference for working with CW complexes instead of simplicial sets. One of the reasons is that the cellular chain complex of a CW complex is often easier to work ...
1
vote
0answers
107 views

cohomology ring of compact submanifolds of Euclidean spaces

Suppose we have a compact $m$-dimensional submanifold $M$ of $\mathbb{R}^N$ and we want to know the cohomology ring $H^*(M;\mathbb{Z})$. Let $\epsilon>0$ and a $m$-dimensional finite simplicial ...
2
votes
1answer
136 views

Basic technical things about simplicial sets to have a good understanding of quasicategories

May someone provide me the list of basic techniques about simplicial sets, in order to have a good understanding of the definition of a quasicategories, $\infty$-topos, $\infty$-stacks, ...
3
votes
0answers
128 views

Test categories applied to Dold-Kan correspondence?

Let's see how this goes, this might be a bit rushed, if you spot any mistakes feel free to correct them. A test category $X$ is a category that can be used in place of the simplex category $\Delta$ to ...
6
votes
0answers
183 views

Grothendieck - A group as a sheaf over simplicial complexes

In this blog post, Terence Tao gives the following definition of a group. Definition. A group is (identifiable with) a (set-valued) sheaf on the category of simplicial complexes such that the ...
6
votes
2answers
356 views

Is this almost-cosimplicial object familiar?

I have a sequence $X_0,X_1,\ldots$ of Abelian groups, along with face maps $d_0,\ldots, d_n: X_{n-1}\to X_n$ which satisfy the standard cosimplicial identities EXCEPT at the very bottom, where in ...
6
votes
1answer
361 views

P.G.Goerss, J.F.Jardine, “Simplicial Homotopy Theory” prerequisites

I know that such questions may be better suited for math.stackexchange, but I believe that that the topic of simplicial homotopy theory is advanced enough for mathoverflow. Besides, I know that there ...
10
votes
1answer
511 views

Categorical or simplicial introduction to modern homotopy theory

I've heard some people saying that it would be easier to study homotopy theory first through simplicial sets. First, i thought that these people were not being serious. But it caught me wondering... ...
7
votes
1answer
522 views

Generalize $\pi_0(B\mathcal{C})\cong\{\text{objects}\}/\{\text{morphisms}\}$ to categories internal to topological spaces

Warmup: Let $\mathcal{C}$ be an ordinary category. Denote by $$B\mathcal{C}=(\coprod_{i\in\mathbb{N_0}}N_{i}(\mathcal{C})\times\Delta^i)/\tilde{}$$ its classifying space, i.e. the geometric ...
3
votes
0answers
103 views

Symmetric sub-simplicial sets of the nerve of $\mathbb{N}$

The nerve of $\mathbb{N}$ is the simplicial set $N\mathbb{N}$ with simplices: tuples $(k_1,\ldots, k_r)$ with each $k_i\in \mathbb{N}$ degeneracies: inserting $0$ faces: adding consecutive entries ...
7
votes
0answers
67 views

Extending a left fibration along an inner horn

Let $\Lambda^n_i \subseteq \Delta^n$ be an inner horn, and let $X \rightarrow \Lambda^n_i$ be a left fibration. Does there exist a left fibration $Y \rightarrow \Delta^n$ such that $X = Y ...
1
vote
0answers
65 views

Cofibrancy of simplicial objects [duplicate]

Let $\mathcal{C}$ be a site. Consider $sPsh(\mathcal{C})$ be the equipped with the local projective model structure. Let $C_{\bullet}$ be a cofibrant object in $\mathcal{C}$ and let $y(C_\bullet)$ be ...
13
votes
3answers
955 views

Does the classification diagram localize a category with weak equivalences?

Let $(C,W)$ be a category equipped with a subcategory of weak equivalences. Its "classification diagram" or "bisimplicial nerve" $N(C,W)$ is a bisimplicial set, for which $N(C,W)_n$ is the nerve of ...
3
votes
1answer
79 views

explicit description of the cosimplicial simplicial set $Q^{\bullet}$

I'm struggling to understand the explicit description of the cosimplicial simplicial set $Q^{\bullet}$ on page 76 (section 2.2.2) of Lurie's book Higher Topos Theory, and would be grateful if someone ...
5
votes
2answers
179 views

What is this construction using iterated face maps of semisimplicial sets?

Let $X$ be a semisimplicial set (face maps but no degeneracy maps). Fix a positive integer $k$. Let $Y_n$ be $X_{(n+1)k}$ and then define $\partial^Y_i:Y_n\to Y_{n-1}$ by $$\partial^Y_i = ...
3
votes
1answer
242 views

Polynomial differential forms on $BG$

Let $\Omega^{*}_{\text{poly}}\: : \: sSet\to dg_{\geq 0}Comm_{+}$ be the polynomial De Rahm functor on simplicial sets, where the codomain is the category of commutative differential graded algebras ...
6
votes
0answers
259 views

Item (4) in Lurie's definition of the class of marked anodyne morphisms

I have a question concerning Remark 3.1.1.3 in Lurie's "Higher Topos Theory" about the definition of the class of marked anodyne morphisms. There, it is mentioned that "it suffices to allow $K$ to ...
7
votes
1answer
254 views

Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar setting)

Given a fibration $p : Y \to X$ in simplicial sets (or any other model category), there are various ways to construct its fibrewise suspension, i.e. its suspension as an object of the slice ...
2
votes
2answers
1k views

The Area of Spherical Polygons

I am interested in finding a canonical general expression for the area of a spherical polygon in $\mathbb{S}^2$ knowing the side lengths of the polygon and a bound on the internal angles (we can ...
5
votes
1answer
212 views

Basic questions about simplicial commutative rings

I am trying to learn about simplicial commutative rings, and would be grateful if one can help with some basic facts about them. Basically, I would like to understand how to do homological algebra ...
5
votes
1answer
122 views

For a simplicial set $X$, is the category of non-degenerate simplices of $X$ a full subcategory of the category of simplices of $X$?

I'm slightly confused, but I think you can help me. Let $X$ be a simplicial set. The category of simplices of $X$ and its subcategory of non-degenerate simplices are defined at ...
7
votes
0answers
173 views

$E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral ...
3
votes
1answer
234 views

Does the fat geometric realization take limits to homotopy limits?

I am deeply confused about geometric realizations and finite limits. Suppose that I am working with simplicial sets (I dont need simplicial spaces) that are "good" in the sense of Segal so that I can ...
6
votes
1answer
410 views

Geometric realization of simplicial spaces and finite limits

Let $X_{\bullet}$ be a simplicial space and denote the (non-fat!) geometric realization by $\lvert X_{\bullet} \rvert$. Does this geometric realization of simplicial spaces preserve finite ...
1
vote
1answer
114 views

Does a topological hypercover always have free degeneracies?

This question arises when I am reading Dugger and Isaksen's "Hypercovers in topology". According to Definition 4.1 in that paper, A hypercover of a space $X$ is an augmented simplicial space $U_*\to ...
2
votes
0answers
104 views

Is the dg-nerve functor a Quillen equivalence?

Lurie defines the dg-nerve $N_{dg}(\mathcal{C})$ of a dg-category $\mathcal{C}$ in Higher Algebra Construction 1.3.1.6: for each $n \geq 0$, we define $N_{dg}(\mathcal{C})_n\simeq ...
8
votes
3answers
632 views

When is the projective model structure cartesian? When is the internal hom invariant?

If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category $Fun(D,M)$ where the weak equivalences are the ...
12
votes
1answer
350 views

Verifying that $\epsilon^!$ is indeed the right adjoint of $\epsilon_*$ in the context of algebraic stacks

The question is about the last paragraph of Remark 12.4.3 in the book on algebraic stacks by Laumon and Moret-Bailly. Let $S$ be a (quasi-separated) scheme and let $\mathscr{X}$ be an algebraic stack ...
9
votes
0answers
137 views

Why should we regard $PL(M)$ as a simplicial group?

Let $M$ be a manifold. If $M$ is smooth, it is clear what $\text{Diff}(M)$ should be, namely it should be the set of diffeomorphisms of $M$ equipped with the topology in which a sequence of ...
4
votes
1answer
199 views

Is the simplicial set $(\Delta_3/\partial \Delta_3)^{\Delta_1}$ finite?

A simplicial set is called finite if it has only a finite number of non degenerate simplicies (or equivalently, if its number of $n$-simplicies grows polynomially with $n$). In an answer to a ...
2
votes
1answer
168 views

Equivalent definition of a Kan fibration

It is known (follows for example from Proposition 4.2 of Simplicial Homotopy Theory by Goerss and Jardine) that a Kan-fibration can be defined as a map having the right lifting property with respect ...