# Questions tagged [simplicial-stuff]

For questions about simplicial sets, simplicial (co)algebras and simplicial objects in other categories; geometric realization, Dold-Kan correspondence, simplicial resolutions etc.

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### Density Theorem for $\infty$-Categories (HTT, Lemma 5.1.5.3)

The density theorem in the ordinary category theory asserts that every presheaf on a small category is a colimit of representables in a canonical way. In Lemma 5.1.5.3 of Higher Topos Theory, Lurie ...

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### Is there an analogy between convergence and homotopical triviality?

Is there an analogy (formal or intuitive) between the notions of convergence and contractibility ?
Is the notion of convergence an instance of homotopical triviality in some formalism, or should be ?
...

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### Décalage and the simplicial path object

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\const{const}$Let $[+1]:\Delta\to\Delta$ be the décalage endomorphism sending $n\mapsto n+1$
adding a new minimal element, i.e. $f:n\to m$ is sent to $...

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### Strongly final vertex $Y$ in a simplicial set gives a retraction of $\mathcal{C}^{ \triangleright }$ onto $\mathcal{C}$

I have a question about a proof from Jacob Lurie's Higher Topos Theory (p 45):
Proposition 1.2.12.5. Let $\mathcal{C}$ be a simplicial set. Every strongly final object (see below what means) of
$\...

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### Defining null-homotopy in terms of endofunctors in an arbitrary simplicial category

I am looking for references defining contractible or null-homotopic in terms of endofunctors $\Delta\to\Delta$.
Let $[+1]:\Delta\to\Delta, n\mapsto n+1$ be the endofunctor of $\Delta$ adding a new ...

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### Classifying the endofunctors of the category $\Delta$ of finite linear orders

Is there a theory of endofunctors of the category $\Delta$ of finite linear orders ?
Can they be classified ? Is there a reference on this ?
Can one classify endofunctors $T:\Delta\to\Delta$ which ...

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### Defining homotopy via the “doubling” endofunctor of a simplicial category

I am looking for a reference explicitly defining simplicial homotopy
in terms of endofunctors of $\Delta$, and developing homotopy theory in this terms.
The following is a particular question.
Is it ...

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### Defining homotopy via endofunctors of a simplicial category

$\newcommand\sSets{\text{sSets}}\newcommand\sing{\text{sing}}\DeclareMathOperator\Hom{Hom}\newcommand\Top{\text{Top}}$I am looking for a reference describing the notions of homotopy and ...

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### Is every simplicial map $\Phi:K(A, n) \to K(A', n)$ a simplicial homomorphism of groups?

I have posted a few questions on MSE, most notably this one, which revolve around the same issue and have received no answers, so I decided to ask the same here.
In the following, $K(A, n)$ is the ...

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### Barycentric subdivision and 1-coskeletalization

Let
$sd : sSet \to sSet$ denote barycentric subdivsion;
$cosk_1 : sSet \to sSet$ denote 1-coskeletalization.
Question: Let $X$ be a graph or simplicial set. If the homotopy type of $cosk_1(X)$ is ...

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### Arbitrary-dimensional expanders?

Rephrasing expansion (slightly). Consider the following slightly tweaked version of the usual definition of a (spectral) expander graph.
(We write a weighted graph as $(V,\beta)$, where the weight $\...

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### Is there an interesting model structure on sSet whose fibered objects are exactly contractible Kan Complexes?

For the Kan-Quillen model structure, the fibered objects are exactly the Kan Complexes and for the Joyal model structure, the fibered objects are exactly the $\infty$-categories. This follows from the ...

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### Homotopy coherent nerve versus simplicial nerve

Background
Recently I asked a question on a particular construction ot the classifying space of a topological group. I got an answer, but it relied on nontrivial Quillen equivalences between various ...

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### Simplicial nerve of a topological group

Given a group $G$, one way to construct the classifying (topological) space $BG$ is to regard $G$ as a category with one object and morphisms $G$, take its nerve, and then apply the geometric ...

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### Homotopy limits indexed by a covering

We all know that the homotopy fiber of a continuous function $f: A \to B$ has a simple expression in terms of points and paths connecting them, that is
$$ \textrm{hofib}_y(f) = \{ (x,\gamma) \in A \...

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### What is the name for the construction of this poset related to coherence of degeneracies of the simplex category?

I present you a family of posets here. I don't say the posets themselves have a conventional name. However I'm sure the general construction of this kind has received some terminology, related to ...

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### Simplicial basis in iterated bar construction

Let $G$ be an abelian group and set $A:=\mathbb{Z}G$. We can define a commutative dga Hopf algebra
$$B(A):=\bigoplus_{k \in \mathbb{Z}}\,B_k,$$
where $B_k:=A^{\otimes(k+1)}$. I like to think of the ...

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### Composition in $\infty$-Categories

This is similar to another question on MO, but is different.
Let $p:\mathcal{E}\to\mathcal{C}$ be a right fibration between $\infty$-categories. As explained in Lurie's HTT, $\S$2.1, we can consider $...

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### Non-equivalent spaces with the same homotopy groups

It is well known that two topological spaces that have all homotopy groups isomorphic need not be weakly homotopy equivalent, because it might not be possible to construct a single map inducing all ...

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### Almost acyclicity of the complex of configuration spaces of noncollinear points in projective plane over finite fields

Let $F$ be a finite field with many elements, say more than 7 for example, and $X$ be the corresponding projective plane $\mathbb{P}^2(F)$.
For a set of points in $X$, if any three of them are ...

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### What is the average degree of a d-simplex?

I am a beginner in network topology topics and while I was reading an article about simplicial complexes where the authors had used random simplicial complexes, I came across a formula using "...

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### Defining Hochschild homology of non-commutative DG-algebras with animated rings with a circle action

A cool construction of Hochschild homology (that I saw on B. Antieau's website here ) is the following:
Let $k$ be a commutative ring, then denote by $\mathfrak{a}\text{CAlg}_k$ the category of ...

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### Do we really need degeneracies for spectral sequence of homotopy simplicial chain complex?

Let's consider an homotopy simplicial chain complex, that is a functor of $\infty$-categories $X_{\bullet} = \textrm{N}(\Delta^{op}) \to \mathcal{C}_{\ge 0}$, where $\mathcal{C}_{\ge 0}$ is the $\...

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### Constructing sections of a cocartesian fibration

Suppose $\mathcal{E} \to \mathcal{C}$ is a cocartesian fibration over (the nerve of) a classical category, and there is a section on zero simplices that sends $C$ to $s(C)$ such that, for every edge $...

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### Cohomology of bisimplicial set is the cohomology of the total simplicial set?

Given a bisimplicial manifold (or set, topological space) there is the bar construction, which assigns to it a simplicial manifold, called the total simplicial manifold.
Now stepping back for a moment,...

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### What does the cotangent complex tell you when it takes animated inputs?

These two links: What is the cotangent complex good for? and Intuition about the cotangent complex? are quite helpful in giving intution for the cotangent complex in terms of deformations but I don't ...

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### The exactness of the associated chain complex of a simplicial free abelian group over a finite set and the normalization theorem

Update:
Now I know why my method fails. But I still wanna know how to work out the original question, that is to show the exactness of the chain complex $C_*(X)$ except for two positions $n=0,N-1$.
...

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### Singular complex and homotopy coherent nerve as simplicial sets

Let $X$ be a CW complex.
Is the simplicial set $\ \mathrm{Sing}\ X$ isomorphic to the homotopy coherent nerve of some Kan enriched category?
Is this true for $X$ = the real line?

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### Is the category of all topological spaces, including the bad ones, simplicially tensored and cotensored?

Let $\textbf{Top}$ be the category of all topological spaces, including the bad ones.
We can make $\textbf{Top}$ into a simplicially enriched category as follows:
Given topological spaces $X$ and $Y$,...

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### Are there local maps of simplicial (co-)cycles on $d$-manifolds beyond cohomology operations?

I'm interested in locally defined maps of cocycles/chains on manifolds of a fixed dimension $d$ which are compatible with cohomology. To be concrete about what "local" means, let me consider ...

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### Does the (1-)topos structure on simplicial sets have any homotopy-theoretic significance?

To give an example of a peculiar feature of simplicial sets that I cannot remember encountering anywhere in the context of homotopy theory: every simplicial set $X$ possesses partial map classifier $X\...

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### Kan–Thurston theorem and R-completion

A corollary of the Kan and Thurston theorem states that the space $X$ (path connected) can be recovered, up to homotopy, by applying the fiber-wise $\mathbb{Z}$-completion functor of Bousfield–Kan (...

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### Chain-level representability of simplicial cohomology

There's a simple construction for a classifying space $K(G,1)$ of a finite group $G$ as an infinite simplicial complex consisting of one $i$-simplex for each possible simplicial $1$-cocycle restricted ...

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### When is the fat join a monoidal structure?

This question is about the following general construction.
Definition:
Let $(\mathcal C, \otimes)$ be a cocomplete, monoidal biclosed category whose unit $\ast$ is terminal. Let $I$ be an "...

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### Why are quasi-categories better than simplicial categories?

There are many models for $(\infty,1)$-categories: simplicial categories, Segal categories, complete Segal spaces, and quasi-categories.
Doubtlessly the model most used to do higher category theory in ...

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### How do the various homotopy 2-categories compare?

There are various models of $\infty$-categories floating around, so there are as many models of the associated homotopy 1- and 2-categories. Because the relations between the former are worked out in ...

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### Formally étale maps of animated $k$-algebras

In Lurie's DAG, he defines what it means for a natural transformation $T:\mathcal{F}\to\mathcal{F}'$ of functors $\mathcal{F},\mathcal{F}':\mathcal{SCR}\to\mathcal{S}$ to be formally étale. Namely, it ...

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### Is there any research about secant varieties by using homotopical algebra or simplicial methods?

Let $A$ be a finitely generated $\mathbb{C}$-algebra with $proj A$ is a projective variety $X \subset \mathbb{P}^n$. The join $J(X,X)$ can be represented by $proj (A\otimes_kA)$ and also we can ...

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### Reference request - Two notions of suspension /loop space agree for simplicial objects

In Goerss-Jardine they point out that there are two reasonable definitions of the (reduced) suspension of a simplicial set. One is the smash product with $S^1$ and one is the join with $S^0$, the &...

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### Example of a non-$\infty$-category whose homotopy category is a groupoid

What is an example of a simplicial set $S$ such that its homotopy category $hS$ is a groupoid, but such that $S$ is not an $\infty$-category?
I know that if $S$ is an $\infty$-category, then $S$ is a ...

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### Simplicial spaces and reflexive coequalisers

Let $X_\bullet$ be a simplicial space. Consider the reflexive coequaliser of $X_1\rightrightarrows X_0$, which we call $X$. Then we clearly have a map $\varphi\colon |X_\bullet|\to X$, where $|X_\...

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### The localization of the span category

Suppose one has a model category $C$ with its class of weak equivalences $W$. It is possible to form a separate homotopical category $(C_{\operatorname{span}},W_{\operatorname{span}})$ which has ...

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### Explicitly calculating the homotopy fiber of a 3-cocycle in the category of simplicial sets

I am stuck at a critical step in my master's thesis. If someome can help me out here, I will give appropriate credit.
We know that the data of a 2-group $G$ can be given by a group $\tau_0(G)$ and a 3-...

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### Algebraic topology and homotopy theory with simplicial sets instead of topological spaces

To quote Kerodon:
In fact, it is possible to develop the theory of algebraic topology in entirely combinatorial terms, using simplicial sets as surrogates for topological spaces.
A similar quote can ...

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### Simplicial polytope with regular cones

Let $P$ be a convex simplicial polytope in $\mathbb{R}^n$. Can we find a convex simplicial polytope $P_0$ in $\mathbb{R}^n$ combinatorially equivalent to $P$, satisfying the following condition: The ...

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### Geometric realisation of smooth $\infty$-stacks

Let $Sh^\infty(\mathsf{Man})$ denote the $\infty$-category of sheaves of $\infty$-groupoids over the site $\mathsf{Man}$ of smooth manifolds (if you prefer, that's the model category of simplicial ...

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### Kan Complexes, proof of extension of a map to a product

I'm reading a book about Kan Complexes in Simplicial Homotopy Theory (Curtis), and came across this theorem at the beginning : if $f : \Delta[n] \times \Lambda^k[m] \to K$ is a simplicial map and $K$ ...

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### Proving that the simplex category is generated by the face and generacy maps

Note: This is not intended to be a research level question, but concerns graduate level material.
Theorem. The opposite $\Delta^\mathrm{op}$ of the simplex category $\Delta^\mathrm{op}$ (as usually ...

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### Can homotopy limits of simplicial sheaves be calculated (correctly) using sheaves of Kan complexes?

$\DeclareMathOperator\holim{holim}$
Let $sSh$ be the category of simplicial sheaves on some site (I like using the psychological crutch of the site having enough points; further, to clarify a bit, &...

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### Why is this class of right-anodyne maps closed under pushouts?

Let $S$ be the class of all right-anodyne maps $r$ such that the pullback of $r$ along any left fibration is again right-anodyne. According to Land's book on $\infty$-categories (specifically the ...