The simplicial-stuff tag has no wiki summary.

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### What are simplicial topological spaces intuitively?

(This is a repost of a question from MSE. I hope there is more to say.)
I tend to imagine simplicial objects in a category as some kind of "topological objects", with a notion of homotopy. Simplicial ...

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**1**answer

227 views

### Thomason’s homotopy colimit theorem for pseudo-functor

Thomason’s homotopy colimit theorem
R W Thomason, Homotopy colimits in the category of small categories,
Math. Proc. Camb. Phil. Soc. (1) 85 (1979)91-109
says that for a functor $F:I^{op}\to ...

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**0**answers

56 views

### Relationship of height zero hypercovers to co-cartesian condition on cosimplicial modules

Suppose given a cosimplicial ring $R^\bullet$ and a cosimplicial module $M^\bullet$ (i.e. a cosimplicial Abelian group such that $M^n$ is an $R^n$-(left/right/bi)module). I have seen it said that ...

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141 views

### Uniformly sampling from the set of all simplicial maps

Let $K$ and $L$ be finite simplicial complexes that remain fixed throughout.
How does one efficiently sample (according to the uniform distribution) elements from the finite set of simplicial ...

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94 views

### Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the cosimplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...

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107 views

### Does compactly supported cohomology make sense for cosimplicial spaces?

As far as I understand, whenever one has something (co)simplicial in spaces, one should take a sort of diagonal to reduce the study to a single space. I'm never sure though whether one preserves only ...

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**1**answer

335 views

### Morita theorem for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting?
I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...

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104 views

### Weak notions of Kan fibrations

I have two semi-simplicial complexes $X_\bullet$ and $Y_\bullet$, along with a simplicial map $f_\bullet: X_\bullet \to Y_\bullet$. Now $f_\bullet$ is not a Kan fibration: for an $n$-simplex in ...

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**2**answers

328 views

### Finite simplicial sets

A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. It is not hard to show that every finite simplicial set has only a finite number of simplicies in ...

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**1**answer

290 views

### Exponentiation in finite simplicial sets

A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. My question is: if $A$ and $B$ are finite simplicial sets, does this imply that the simplicial set ...

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560 views

### Is every connected space equivalent to some B(Aut(X))?

Given a connected space $B$, is there always some space $X$ with $B \simeq \mathbf{B}(\mathrm{Aut}(X))$?
Here by space I mean simplicial set, by $\mathrm{Aut}(X)$ I mean the simplicial monoid of ...

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**1**answer

161 views

### What is the suitable setting for supercoherence with value in a bicategory?

It was J.F. Jardine established the so called supercoherence theory in Journal of Pure and Applied Algebra Volume 75, Issue 2, 18 October 1991, Pages 103–194. The result can be roughly stated as ...

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**1**answer

137 views

### On the obstruction of a sequence of simplicial spaces that is levelwise a fibration

Given a sequence of simplicial spaces (actually bisimplicial sets)
$$F\to E\to B$$
that is level-wise a fibration, then the geometric realisation does not necessarily have to be a fibration.
If I ...

**3**

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**1**answer

441 views

### Projective objects in HTT

In HTT.5.5.8.18 Lurie defines a projective object $P$ in a quasicategory $\bf C$ as an object such that its corepresented functor ${\rm Map}(P,-)$ "commutes with geometric realizations". I can catch ...

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373 views

### Do Homotopy Fully Faithful Functors Push-out?

A (homotopy) fully faithful functor is a map of $\infty$-categories which induces weak equivalences on mapping spaces.
Are homotopy fully faithful functors preserved under (homotopy) pushout?
More ...

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**1**answer

248 views

### Two definitions of modules in monoidal category

The standard definition of a (left) simplicial module $V$ over some simplicial algebra $A$ is the map of simplicial vector spaces $A\otimes V\to V$ that gives the usual modules component-wise. Here ...

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105 views

### What is the left adjoint for taking rows of a bisimplicial set?

Let $aSS$ / $abSS$ be the category of augmented bi/simplicial sets (one can also consider $SS$/$bSS$ be the usual bi/simplicial sets, the results should related in some reasonable way.)
There is an ...

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341 views

### Simplicial replacements in smoothing theory

As far as I can tell, ever since Milnor's Microbundles and differentiable structures (1961) paper, whenever people talk about $Diff(\mathbb R^n)$ or $PL(\mathbb R^n)$ or $Homeo(\mathbb R^n)$, they ...

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**1**answer

91 views

### Poset complex of reverse ordering [closed]

This might be too easy but I cannot proof it easily. Any reference or hint will be great.
Q: Suppose P is a poset in which every chain is finite and $\Delta P$ is the poset complex associated to it. ...

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278 views

### How to detect if a simplicial set is the nerve of a groupoid?

I have the following question.
Suppose I have a simplicial set. Is there a way to detect if it actually is isomorphic to a nerve of a groupoid?
I've seen the fact that if you have a nerve ...

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177 views

### What does it mean for the $\Pi_\infty$ groupoid to be fully fibrant?

The original question I had was:
If I have a sequence of simplicial spaces
$$A\to B\to C$$
which is degree-wise a homotopy fibration, under which conditions is
the geometric ...

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217 views

### The normalised cochain complex, totalisation and cosimplicial simplicial $R$-modules

Short Version
Given a cosimplicial space $X_\bullet$, what is the relationship between (co)chains on the totalisation of $X_\bullet$ and the totalisation of the cosimplicial chain complex obtained by ...

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178 views

### Finding automorphism groups of simplicial complexes

Question:
Given a finite simplicial complex $K$, what general techniques allow one to efficiently compute (a presentation of) the group $\text{Aut}(K)$ of $K$'s automorphisms?
Since this is ...

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**1**answer

205 views

### What are the morphisms in the category of zig-zags?

For some reason I am having trouble locating a transparent explanation of precisely what are the morphisms in the category of zig-zags. The objects of this category are specified completely by triples ...

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208 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 ...

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**1**answer

273 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 ...

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171 views

### Adding morphisms to a category without changing homotopy type

I have a really tame category $C$: there are only finitely many objects $C_0$, each hom-set $C(x,y)$ for $x,y \in C_0$ has at most one element, and (aside from identity morphisms) if $C(x,y)$ is ...

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262 views

### The nerve of categories preserves weak equivalence?

Does the nerve functor $\operatorname{Cat}\to \operatorname{sSet}$ from small categories to simplicial sets preserve weak equivalences?
If $f\colon C\to D$ and $g\colon D\to C$ are functors of small ...

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285 views

### How does one Segal-subdivide a 2-category?

Let $\mathcal{C}$ be a small category. Then, its Segal subdivision $\text{sd }\mathcal{C}$ is a new category whose objects are morphisms of $\mathcal{C}$, and a morphism from $f:x \to y$ to $g: w \to ...

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131 views

### Where does the ''hyper'' go?

There is a model structure on the category $Grpd$ of groupoids such that weak equivalences are equivalences of categories, cofibrations are the injections on objects (see nlab) and there is a Quillen ...

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235 views

### Is there a class of simplicial sets whose weak homotopy type is preserved by symmetrization?

We have the categories, $S$, of simplicial sets and $SS$, of symmetric simplicial sets (whose simplices are unordered). There are functors:
$H:S\to SS$ forgetting the ordering on simplices and
...

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107 views

### Dold-Kan preserves weak equivalences and fibrations

It's well known that Dold-Kan correspondence is an isomorphism between simplicial vector spaces and non-negative chain complexes of vector spaces. Moreover, weak equivalences and fibrations are ...

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428 views

### Three questions on $\operatorname{hocolim}$

I posted this on math.stack.exchange but didn't get a helpful response, so please let me try it here.
Let $D$ be a small category and $F:D\to sSets$ a functor.
There is a bisimplicial set indicated ...

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378 views

### how to make the category of chain complexes into an $(\infty,1)$-category

Related to this question, I would like to know, if there is an explicit presentation
of the $(\infty,1)$-category of a model category of abelian chain complexes, but this time in terms of simplicial ...

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415 views

### Can we invert barycentric subdivision?

With apologies to fellow algebraic topologists, I confess that I have no idea how to answer this innocent-looking question:
(1) Let's say we know that a finite simplicial complex $S$ is the ...

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**1**answer

155 views

### Vector fields on a simplicial manifold.

Is there a known definition of vector fields on a simplicial manifold?
For me, it seems natural that the definition should be something along the lines: Let $M_{\bullet}$ be a simplicial manifold ...

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132 views

### Explicit Lie May structure on cosimplicial DG Lie algebras

In the paper "Homotopy Lie algebras", Schechtman and Hinich proved that any cosimplicial
differential graded Lie algebra has the structure of a 'Lie May algebra'.
If my understanding is right here, ...

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**1**answer

296 views

### Does every simplicial polytope have a topology-preserving contractible edge?

An edge of a triangulated manifold is said to be contractible if it may be contracted to a vertex without modifying the topological type of the underlying manifold. Otherwise, the edge is ...

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172 views

### Is there a local projective model structure on simplicial sheaves? What are its fibrant objects?

Consider a site S (I am mostly interested in hypercomplete sites, e.g., the site of smooth manifolds).
The category of simplicial presheaves SPSh(S) on S can be equipped with the local projective ...

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326 views

### Removing a simplicial subset from a simplicial set

Let $A, X$ be simplicial sets, and suppose there's an inclusion $A \longrightarrow X$. Geometrically realizing the inclusion map, we get a pair of spaces $(\mathcal{A}, \mathcal{X})$. I want to find ...

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222 views

### Simplicial chain complex with ordered simplices

Let $X$ be an abstract simplicial complex. Recall that the usual simplicial chain complex for $X$ is defined as follows. Let $C_k(X)$ be the quotient of the free abelian group on formal symbols ...

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398 views

### Combinatorial distance between simplicial complexes

Let $K_1$ and $K_2$ be two simplicial complexes.
I am seeking a measure of the distance between $K_1$ and $K_2$ when
viewed as combinatorial objects.
What I have in mind is something like this.
...

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**1**answer

167 views

### How to call a simplicial set where every boundary of a simplex can be filled?

What is the correct terminology for the following property of a simplicial set $X_\bullet$:
For every $k\geq 0$, every map $\partial\Delta^k\to X_\bullet$ can be extended to a map $\Delta^k\to ...

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**1**answer

116 views

### homology of $B S^{-1} S$ computation in the proof that $+ = Q$

Let $S$ denote the category of projective (left) $R$-modules with isomorphisms for arrows. We have that
$BS^{-1}S \sim B \text{GL}(R)^+ \times K_0(R)$
In proving this, in Srinivas' algebraic ...

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227 views

### Path components of a monoidal category acting on homology

Let $S$ be a (small) symmetric monoidal category and $X$ a (small) category on which $S$ acts. $\pi_0(S) = \pi_0(BS)$ is naturally an abelian monoid, with $[A] + [B] := [A+B]$, where $[A]$ denotes ...

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575 views

### Multisimplicial geometric realization

Does anyone know a reference or proof for the following? Let $k\geq 1$ and let $X$ be a space. There is a $k$-fold multisimplicial set whose simplices in degree $n_1,\ldots,n_k$ are the maps ...

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233 views

### Simplicial space whose all face/degeneracy maps are homotopy equivalences

I believe that the following is true, but I cannot find a proof. Let $X_\bullet$ be a simplicial topological space (I can add that my $X_\bullet$ comes from a bisimplicial set, so the spaces $X_n$ are ...

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471 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 ...

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427 views

### Simplicial path and loop spaces

I am trying to understand the relationship between the simplicial path space and loop space with the path space of a topological space, and the loop space of a topological space.
I have understood ...

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### realization of maps between classifying spaces of categories

The classifying space $B\mathcal{C}$ of a small category $\mathcal{C}$ is by definition the geometric realization of the nerve of $\mathcal{C}$. Now let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two ...