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### Equalizer in Free groups

Let $F_n$ be the free group generated by $x_1,\cdots,x_n$ and for each $1\leq i\leq n$, let a homomorphism $d_i:F_n\to F_{n-1}$ be defined as follows:
$d_i(x_r)=x_r$, if $i>r$;
$d_i(x_r)=1$, if ...

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### Does the nerve functor preserve fibrations?

As asked in the title but more specifically: does the nerve functor from Cat to sSet map a fibration between groupoids to a Kan fibration ?
By fibration of groupoids I mean a fibration for the ...

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### Is the discrete nerve of a small category a complete Segal space?

While reading Rezk's paper "A model for the homotopy theory of homotopy theory", I found a remark which contradicts a guess of mine, but I can't see where I am wrong (perhaps it might be a silly ...

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### contracting homotopy on simplicial sets

Let $X$ be a topological space and let $PX$ be its space of paths. Let $I=[0,1]$ with coordinate $s$. There is an homotopy
$$
F\: : \: I\times PM\to PM
$$
Defined by $F(s,y)(t):=y(st)$. This map is an ...

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### To what extent are homotopy colimits over a weakly contractible category “determined by local data”?

[I'd be very happy for a better question title, if anyone has any suggestions.]
I have a category $C$, two functors $F,G : C \to \mbox{Cat}$, a natural transformation $\alpha : F \to G$, and a ...

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119 views

### Clarification about Joyal's notation [closed]

At the very beginning of chapter 5 of Joyal's lectures on Quasi-Categories (http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), he uses a notation which I think he has never ...

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### When do zero-simplices of a simplicial diagram determine its homotopy colimit?

Suppose that I have a diagram of simplicial sets $X_\bullet:\mathscr{C} \to Set^{\Delta^{op}},$ with $\mathscr{C}$ a small category such that for each $C \in \mathscr{C},$ $X_\bullet(C)$ is a Kan ...

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### Cell(J) vs Cof(J) in $\text{sSet}_{\text{Quillen}}$

consider sSet equipped with its Quillen model structure $\text{sSet}_{\text{Quillen}}$, we know that a trivial cofibration is a retract of a transfinite composition of pushouts of horn inclusions. I ...

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### When is a topological space the homotopy colimit of an open covering?

Suppose that $X$ is a topological space and $\left(U_i \to X\right)$ is an open cover. We can associate to it the Cech diagram of this cover $$C_U:\Delta^{op} \to Top.$$ I know that for many good ...

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### Trying to understand straightening functor associated to a right fibration of simplicial sets

Let $p:X \to S$ be a right fibration of simplicial sets; one can roughly think of it as some sort of "functor" $S^{op} \to Set_\Delta$ (where $Set_\Delta$ denotes simplicial sets) sending $s \in S$ ...

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### Freely adding degeneracies does not change the homotopy type

Background: Let $S$ be a simplicial set. By freely adding degeneracies to $S$, I mean first applying the forgetfull functor from simplicial sets to semi-simplicial sets which forget the already ...

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### What suffices to check completeness in an n-fold Segal space?

Recall that a Segal space is a simplicial space $X : \Delta \to \mathrm{Spaces}$, $\bullet \mapsto X_\bullet$, which satisfies the
Segal condition: For each $j$, the map
$$ X_j \to ...

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345 views

### Higher Cerf Theory

Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that ...

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### When is the product of an infinite family of simplicial sets also a homotopy product?

The homotopy product of an infinite family of simplicial sets can be computed
by deriving the product functor sSetW→sSet, for example,
by performing the componentwise fibrant replacement using Kan's ...

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### Why $D^b(S)\cong D^b_{\text{Car}}(\text{cosq}(X\rightarrow S))$?

Let $p: X\rightarrow S$ be a map between topological spaces and we can construct the simplicial space $\text{cosq}(X\rightarrow S)$ where $X_0=X$, $X_1=X\times_S X$ and
$$
X_n=\underbrace{ X\times_S ...

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### 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 ...

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### Recognizing Simplicial (Quasi)Fibrations

Let's say we are given two finite simplicial complexes, which I will suggestively call $E$ and $B$. We'd like an algorithm for the following decision problem:
Does there exist a simplicial map ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### Pseudo-manifolds and homology

Is there a good reference for the proof that the cobordism group of pseudo-manifolds is isomorphic to the singular homology group?
I was looking for a more geometrical definition of homology and ...

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### A model structure on marked simplicial sets

Do you have a reference for the following fact? And before that, is it true?
The Joyal model structure on simplicial sets "lifts" to a model structure on the category of marked simplicial sets, ...

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177 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 ...

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### 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. ...

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### 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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### 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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### 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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### When does a sheaf of categories represent a homotopy sheaf?

Suppose that $F$ is a sheaf of categories (on a Grothendieck site or even a topological space). By this, I mean a sheaf in the naive 1-categorical sense, so it can equivalently be viewed as a category ...

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### Why does the singular simplicial space geometrically realize to the original space?

I have seen it claimed that (for compactly generated Hausdorff spaces) the geometric realization of the singular (internal) simplicial space is homotopy equivalent to the original space. I know how to ...

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### 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 ...

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### 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 ...

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### A natural simplicial object in the simplicial category (?)

In several works (es. [CS]) the study of the properties of the simplicial category $\Delta$ reveals fundamental aspects of universal properties (eg monoid generator) or basic constructions (eg ...

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### References for the “nerve of an algebraic variety”

Let's do algebraic geometry over an arbitrary base ring $k$.
I've frequently seen a definition of the algebraic $n$-simplex, as follows:
$$\Delta^n = ...

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### 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 $\mathbf{t}$ ...

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### 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 ...

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### Homotopy theory of suplattices

In Quillen's monograph Homotopical algebra where he introduced the notion of model category, he showed that if $C$ is a bicomplete category with enough regular-projectives in which either (*) every ...

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### Has this chain complex associated with a simplicial complex been studied before?

I have stumbled upon a construction that has probably been noticed before, and I wonder if anyone can point me to a reference.
Suppose that $K$ is a simplicial complex. Let $P(K)$ be the free abelian ...

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### 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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### Recognition principle for 2-categories (2-groupoids)

Given a 2-category (i.e. bicategory) $C$ there is a unitary geometrical nerve whose 0-simplices are objects of $C$, 1-simplices are 1-arrows of $C$, 2-simplices are 2-commutative triangles (in certain ...

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### A question about the proof of Quillen's Theorem A

(I posted this question on Mathstack but I haven't received any answers or comments so I thought I might as well try my luck here. I apologize if it is not an appropriate question.)
Theorem (Quillen) ...

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### Prove the functor $[n]\to [n]\star [n]$ preserves inner anodyne

Let $f:\Delta\to \Delta$ be the functor given by $[n]\mapsto [n]\star[n]=[2n+1]$. We can extend $f$ cocontinuously to a functor
$$f_!: SSet\to SSet$$
(that is, the left adjoint of the functor $f^*$. ...

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### Is there any relation between the simplicial $S^1$ and the Hochschild homology of a noncommutative algebras

Let $k$ be the base field and $A$ be a unital associative $k$-algebra. Let's review the Hochschild homology theory: we have the Hochschild chain comple $C_{\cdot}(A)$ where
$$
C_n(A):=A^{\otimes n+1}
...

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### Group cohomology without G-modules (a.k.a. what does this bar construction compute?)

Without any prior exposure to the cohomology of groups, one might naively proceed by replacing a group by a sort of resolution.
For instance, let's take $G = \mathbb{Z}^2$, and "resolve":
$$ 0 \to ...

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### Cocycle condition for equivariant sheaves

Let $G$ be an affine group that acts on a variety $X$. Equivariant sheaves on $X$ could be defined in the following way. Consider the simplicial space $X_\bullet$
: $X_n := G^n \times X$, $s_0:X_0 ...

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### How to define $\Lambda^0_0$, 0-horn of a simplicial point

This is really a trivial question.
The 0-horn of a simplicial point $\Delta^0$ is not defined nor remarked in the books and papers I could find. So one expect if we could make some meaningful ...

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### 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 ...

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### 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 ...

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### The bar construction or the quotient for monoidal category action on a category

Given a monoid $C$ acting (from the right) on a set $M$, there is a bar construction giving a simplicial set or equivalently a translation/action category,
$$
N_\bullet (M\rtimes C)= \cdots M\times ...

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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 ...