# Tagged Questions

**3**

votes

**0**answers

126 views

### Getting a Postnikov Tower from the Tot-tower?

I have had a problem I've been thinking of recently but can't seem to make anything of it.
Let $X$ be a simplicial set. It is well known that one can construct a Postnikov tower
$$P_nX \rightarrow ...

**2**

votes

**0**answers

63 views

### Why does $\pi_t$ preserve pullbacks in this special case?

Let $X$ be a fibrant pointed cosimplicial space.
Following Bousfield-Kan, let $\text{lim}^{\partial \Delta_{n+1}} X = M^{n}X$ be the nth matching object of $X$. Onecan then show that there is a ...

**4**

votes

**1**answer

116 views

### In a Simplicial group the Degenerate elements don't intersect the kernel of the face maps.

Let $G_{\bullet}$ be a simplical group. I denote by $D_n \subset G_n$ the subgroup generated by all the degeneracy maps $s_i:G_{n-1} \to G_n$.
I also denote $$M_n = \bigcap_{i>0} \mathrm{Ker} d_i ...

**0**

votes

**0**answers

48 views

### Homotopy addition theorem and lifting in a certain diagram

If I am confusing somewhere here, please excuse me and ask me to clarify. The proof I am having trouble with is lemma 1.19 of Goerss-Jardine, pg. 398. I have tried to isolate the troublesome ...

**2**

votes

**1**answer

85 views

### Does the right adjoint of the category of simplices functor is “homotopicaly inverse” to the category of simplices functor?

Short Version (the question)
Let $\text{Cat}$ be the category of (small) categories and $\text{sSet}$ the category of simplicial sets. There is a functor $\Gamma:\text{sSet}\to \text{Cat}$ that takes ...

**1**

vote

**1**answer

165 views

### Milnor's exact sequence and a certain proof

Please forgive me if this is not the right forum for this question.
Let $$ X = \cdots \rightarrow X_n \rightarrow X_{n-1} \rightarrow \cdots \rightarrow X_0 = \ast$$ be a tower of fibrations of ...

**4**

votes

**0**answers

69 views

### Is a finite-dimensional simplicial set the homotopy colimit over a truncated simplicial category?

If $K$ is a simplicial set, we can extend it to a bisimplicial set $K^\prime: \Delta^{op} \to sSet$ by setting $K^\prime_n = (K_n)_\delta$ where $(-)_\delta$ means taking a discrete simplicial set. ...

**1**

vote

**3**answers

223 views

### Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?

Let $\pi _1:SS\to Grpd$ denote the fundamental groupoid functor, from simplicial sets to groupoids, and let $N:Grpd\to SS$ denote the nerve functor. Then $\pi _1$ is left adjoint to $N.$
On ...

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vote

**0**answers

63 views

### When can I compute the simplicial mapping space from a presheaf to a simplicial presheaf naively?

Suppose that $\mathscr{C}$ is is a small category, $Y$ is a presheaf (of sets) on $\mathscr{C},$ and $X_\bullet$ is a simplicial presheaf. There is a spectrum of simplicial model structures on ...

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votes

**2**answers

137 views

### Mapping complexes in the simplicial localization of the category of manifolds

Let $\mathit{Mfd}$ denote the category of smooth manifolds. Let $W$ denote all projections of the form $$M \times \mathbb{R} \to M.$$ Let $\mathit{Mfd}_W$ denote the Hammock localization of ...

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votes

**2**answers

374 views

### Is the geometric realization of a level-wise weak equivalence a weak equivalence?

For the purposes of this question a topological space will mean a compactly generated weak Hausdorff space, though I am actually somewhat flexible on what category of topological spaces we use. I ...

**2**

votes

**0**answers

82 views

### How to understand $\mathcal{L}BG \simeq G/^{\text{ad}}G$ in term of simplicial sets?

First let $G$ be a topological group and $BG$ its classifying space. Let $\mathcal{L}BG=\text{Map}(S^1, BG)$ be the free loop space of $BG$.
We can see that $\mathcal{L}BG$ has the homotopy type of ...

**2**

votes

**1**answer

171 views

### A certain kind of simplicial complex

I'm interested in collections $\mathcal{C}$ of tuples $\mathbf{t} = (n_1, n_2, \ldots, n_r)$ of positive integers satsifying
if $\mathbf{t}\in \mathcal{C}$ then so is any permutation of $\mathbf{t}$ ...

**6**

votes

**1**answer

218 views

### Are there models for homotopy colimits and limits of simplicial sets that generalize Kan's suspension and loop functors?

Consider the category C of pointed simplicial sets.
The pair of functors X∈C↦X∧S¹∈C and Y∈C↦Map(S¹,Y)∈C
models the suspension and loop functors on the underlying ∞-category of C.
There is another ...

**8**

votes

**1**answer

306 views

### Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid

Let $C$ be a category with an object $X$ such that there are no non-trivial endomorphisms $X\rightarrow X$. Consider a simplex $\sigma$ of the nerve $NC$ of $C$. It is just a string of composable ...

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votes

**1**answer

233 views

### Equivariant homotopy, simplicially

It is a classic result of Kan that the homotopy categories (with appropriate model structures) of simplicial sets and of topological spaces (in fact, one could only care about CW-complexes) are ...

**1**

vote

**0**answers

63 views

### What is a homotopy between bisimplicial maps

I originally posted it on stackexchange, but did not get any comments or answer, so I try my luck here.
I am looking for the naive notion of homotopy.
For maps $f, g: A\to B$ of simplicial sets ...

**8**

votes

**1**answer

713 views

### 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 ...

**1**

vote

**1**answer

202 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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vote

**0**answers

104 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 ...

**9**

votes

**1**answer

278 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 ...

**12**

votes

**2**answers

544 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 ...

**5**

votes

**1**answer

134 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 ...

**2**

votes

**3**answers

241 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 ...

**3**

votes

**0**answers

170 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 ...

**5**

votes

**0**answers

185 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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votes

**0**answers

170 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 ...

**5**

votes

**2**answers

278 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 ...

**3**

votes

**0**answers

128 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 ...

**7**

votes

**1**answer

415 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 ...

**2**

votes

**0**answers

130 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, ...

**2**

votes

**1**answer

225 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 ...

**12**

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**2**answers

557 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 ...

**2**

votes

**2**answers

222 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 ...

**7**

votes

**3**answers

434 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 ...

**6**

votes

**2**answers

278 views

### 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 ...

**7**

votes

**0**answers

157 views

### Tangent space, metrics etc. on simplicial sets

Is there a way to attach some sensible notion of tangent space to a simplicial set? If yes, is it possible to transfer typical local data from differential geometry such as metrics to this setting?
...

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vote

**0**answers

165 views

### Geometric interpretation of higher simpicial homotopy groupoids.

As a geometric interpretation of the simplicial fundamental groupoid, resp. its equivalent relation I thought about something like arcwise differentiable (or arcwise linear) paths modulo "we can go ...

**4**

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**0**answers

259 views

### How to endow an n-fold Segal Space with a symmetric monoidal structure?

I would like to understand how I can endow an n-fold (complete) Segal Space with a symmetric monoidal structure. My question is basically the same as in this post: What is a symmetric monoidal ...

**1**

vote

**1**answer

93 views

### Does the following categorial sum preserve weak equivalences?

In Dwyer and Kan's 1980 paper on "Simplicial Localizations of Categories", they prove the following result for binary categorial sums. For a set $O$, let $O$-${\mathsf{Cat}}$ be the following ...

**6**

votes

**4**answers

693 views

### Is there a sense in which the homotopy theory of simplicial sets is the “paradigmatic” one?

I could not come up with a better title for my question.
What I am asking is this (preemptive excuses to all experts in homotopy theory for naivetes of all kinds you may find herein):
the ...

**3**

votes

**1**answer

196 views

### Explicit contraction for the universal simplicial bundle WG

For a simplicial group $G$, there is a universal bundle $WG \to \overline{W}G$ in the category of simplicial sets, detailed in for example May's book (djvu).
Now $WG$ has a simple enough description ...

**7**

votes

**1**answer

264 views

### What is the homotopy type of a free simplicial ring?

Is there a good description of the homotopy type of a free simplicial ring (or simplicial $R$-algebra) on a given simplicial set, in terms of the homotopy type of that simplicial set?
(This is mostly ...

**4**

votes

**0**answers

207 views

### good covers and simplicial maps

Let $X$ be a paracompact topological space and choose a good cover $U_i$ of $X$. Remember that a good cover is one that consists of open subsets, such that each set $U_i$ is contractible and all ...

**3**

votes

**2**answers

500 views

### simplicial spaces without degeneracies

Suppose I have a simplicial space $X_{\bullet}$ without degeneracies (sometimes called semi-simplicial space or incomplete simplicial space). There still is a geometric realization $\lVert X \rVert$ ...

**7**

votes

**2**answers

551 views

### How to compute homology of symmetric products of complexes?

First, I would appreciate references on the notion of derived symmetric powers of perfect modules over various kinds of derived commutative algebras (say cdgas in characteristic zero, simplicial ...

**3**

votes

**2**answers

220 views

### When does $M \otimes_{A} \pi_{0}(A) \simeq 0$ imply $M \simeq 0$?

Let $A$ be a simplicial commutative ring over a field $k$ of characteristic zero (or a cdga
in non-positive degrees with differential of degree -1). Let $M$ be a perfect $A$ module. If necessary, ...

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votes

**4**answers

381 views

### (Co)homological characterization of homotopy pullbacks

For a commutative square of spaces (of manifolds, or of simplicial sets):
$$S=\left(\begin{array}{ccc} A & \to & B \newline \downarrow & & \downarrow \newline C & \to & ...

**3**

votes

**1**answer

359 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 ...

**8**

votes

**1**answer

520 views

### Computing homotopy (co)limits in a nice simplicial model category?

I'm trying to learn about and compute homotopy (co)limits. Specifically, if $\mathcal{C}$ is some Grothendieck site and $\mathcal{P}$ the simplicial model category of simplicial presheaves (say with ...