6
votes
1answer
188 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
1answer
298 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 ...
9
votes
1answer
199 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
0answers
61 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
1answer
622 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 ...
0
votes
0answers
130 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 ...
1
vote
0answers
100 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
1answer
260 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 ...
11
votes
2answers
515 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
1answer
125 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
3answers
221 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 ...
4
votes
0answers
164 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
0answers
166 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 ...
4
votes
0answers
164 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
2answers
264 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
0answers
122 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
1answer
365 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
0answers
126 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
1answer
223 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
votes
2answers
530 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
2answers
210 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
3answers
327 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
2answers
274 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
0answers
143 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? ...
1
vote
0answers
164 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
votes
0answers
248 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
1answer
91 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
4answers
682 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
1answer
191 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
1answer
261 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
0answers
199 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
2answers
457 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
2answers
496 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
2answers
212 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, ...
4
votes
4answers
370 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
1answer
335 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 ...
7
votes
1answer
465 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 ...
11
votes
3answers
497 views

Necessity of hypercovers for sheaf condition for simplicial sheaves

I'm trying to understand where the definition of simplicial sheaf on a space/site comes from. For a presheaf $F$ of sets on a topological space $X$, the sheaf condition can be viewed as saying that ...
2
votes
1answer
365 views

Simplicial presheaves enriched, tensored and cotensored over simplicial sets correctly?

Let $\mathcal{C}$ be a category and $\mathcal{P}=Functors(\mathcal{C}^{op},\mathcal{S})$ the category of simplicial presheaves, where $\mathcal{S}=sSet$. I want $\mathcal{P}$ to be enriched, tensored, ...
4
votes
1answer
276 views

Writing an infinity groupoid as a colimit of sets

If we are given a simplicial set $X:\Delta^{op} \to Set$, we may regard it as a \emph{simplicial} simplicial set, i.e. a bisimplicial set by composing with the "constant" inclusion $Set \to ...
8
votes
0answers
390 views

Does the category of strict $2$-categories together with Dwyer-Kan equivalences provide a model for $(\infty,1)$-categories?

The question is the title. In what follows, all $2$-categories and $2$-functors will be strict. Let $2-Cat$ denote the categories whose objects are $2$-categories and whose morphisms are ...
4
votes
0answers
184 views

Is the class of inner-anodyne morphisms right-cancellative with respect to the of the class of monomorphisms?

Recall: Given a category $A$, and two classes of morphisms $S,S'$, we say that $S$ is right-cancellative with respect to $S'$ if for any pair of maps $f\in S, g\in S'$ such that $gf$ is defined, we ...
10
votes
1answer
771 views

The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?

Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying ...
9
votes
1answer
730 views

Is the simplicial completion of a localizer always a bousfield localization of the injective model structure?

Background Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following ...
11
votes
2answers
1k views

What is the homotopy theory of categories?

I've heard that Grothendieck, in his letter "Pursuing Stacks," wanted to find alternative models for the classical homotopy category of CW complexes and continuous maps (up to homotopy), and one of ...
10
votes
4answers
768 views

Degeneracies for semi-simplicial Kan complexes

By a semi-simplicial set I mean a simplicial set without degeneracies. In such a thing we can define horns as usual, and thereby "semi-simplicial Kan complexes" which have a filler for every horn. ...
7
votes
2answers
531 views

Historical Question about Simplicial Sets

I have a pretty easy historical question about simplicial sets. Unless I am mistaken, simplicial sets first came out of topology, explicitly from combinatorial topology and the study of simplicial ...
7
votes
0answers
292 views

A combinatorial approximation functor sSet->qCat

Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap ...
3
votes
2answers
326 views

Model categories of simplicial objects

If $\mathcal{C}$ is a category, then surely the category of simplicial objects $s\mathcal{C}$ is not automatically a model category. What conditions must $\mathcal{C}$ satisfy in order for ...
2
votes
1answer
212 views

Local injective model structure for simplicial presheaves

The category of simplicial presheaves on a small Grothendieck site $\mathcal{C}$ can be given a model structure by defining weak equivalences and cofibrations sectionwise. It's called the (global) ...