# 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.

798
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### How to characterize this condition for commutative squares in $\Delta$

In the simplex category $\Delta$ we have the situation, that
pullbacks exist for cospans $[a] \xrightarrow{\alpha} [n] \xleftarrow{\beta} [b]$ in $\Delta_\text{mono}$ and
pushouts exist spans $[a] \...

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1
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### Is there an error in Lurie, HTT, Proposition 6.1.2.6.?

Let $\mathcal{C}$ be an $\infty$-category (meaning quasi-category) and $U\colon N(\Delta)^\text{op}\rightarrow\mathcal{C}$ a simplicial object of $\mathcal{C}$. For a simplicial set $K$, let $U[K]$ ...

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### Understanding the concept of homotopy fixed points

I apologize in advance if this question is too basic for this site, I tried to search online and through the literature for a few days with no success already.
I am trying to understand the concept of ...

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0
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### Slice category of simplicial presheaves

If $\mathcal{C}$ is a small category and we let $\mathbf{Pre}(\mathcal{C})$ denote the category of presheaves over $\mathcal{C}$, then it is well known that for any presheaf $X \in \mathbf{Pre}(\...

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### Simplicial complexes for measuring semantics of concepcts from natural languages?

In a nice paper describing some of the encodings of the semantics in large language models, simplices occur:
Here is a quote from their abstract:
"We find a remarkably simple structure: simple ...

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0
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### Equivalence of two definitions of relative limits

This is a question on seemingly equivalent definitions of relative limits, formulated in the language of quasi-categories. I will use notations from Higher Topos Theory.
Let $p:\mathcal{C}\to\mathcal{...

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1
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### Nerve theorem for simplicial sets

There are various kinds of nerve theorems. I am wondering if the following version of nerve theorem for simplicial sets is true:
Let $X:\Delta^{\mathrm{op}}\to \mathrm{Set}$ be a simplicial set. Let $\...

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1
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### Are simplicial commutative inverse semigroups fibrant?

Let $X$ be a simplicial object in the category of commutative inverse semigroups (or monoids, if needed). Is the underlying simplicial set of $X$ always a Kan complex? If so, are there some nice ...

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### Simplicial right Kan extensions and Cartesian transformations

I will write the concrete question first, in case the answer is clear independently of the context:
Question: Given an $\infty$-topos $\mathfrak{X}$ and a diagram $F\colon\Delta^1\times\Delta_+^{op}\...

9
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1
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### Is the standard model structure on reduced simplicial sets cofibrantly generated?

Let $\mathrm{sSet}_0$ be the category of simplicial sets with a single zero cell, also known as reduced simplicial sets. It is a well known fact (due to Quillen) that $\mathrm{sSet}_0$ supports a ...

7
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1
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### Why Faonte called "small" and "big" dg-nerves?

I read G.Faonte "Simplicial nerve of an $A_\infty$-category" (https://arxiv.org/abs/1312.2127). In his paper, he calls two dg-nerve construction "small" and "big".
"...

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### Explicit formula for Dold-Kan projection to normalized Moore complex

When proving the classical Dold–Kan correspondence (as e.g. in Goerss-Jardine book or here http://math.uchicago.edu/~amathew/doldkan.pdf), one associates three chain complexes to a simplicial abelian ...

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1
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### Gluing $n$ $2(n-1)$-simplices

It should be true that $n$ $2(n-1)$-simplices can be glued together, such that $k$ of them intersect in a common $(2n -k -1)$-cell and the resulting object is a "convex $2(n-1)$-disc" whose ...

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### Quillen's theorem A/B for simplicially enriched categories

I'm looking for any reference which states some version of Quillen's theorem A/B for simplicially enriched categories (so to be clear, simplicial objects of $\mathrm{Cat}$ whose simplicial set of ...

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### Is the mapping space from $\Delta[k]$ to a simplicial set $X$ weak equivalent to $X$?

Let $X$ be a simplicial set. It is well known that a model for the path object can be given by the mapping space $\mathrm{Hom}(\Delta[1], X)$. In particular this offers a fiber replacement for the ...

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### When do the different notions of homotopy inside a general simplicial set agree?

$\newcommand{\defeq}{\overset{\mathrm{def}}{=}}\newcommand{\rmR}{\mathrm{R}}\newcommand{\rmL}{\mathrm{L}}\newcommand{\End}{\mathrm{End}}\newcommand{\Hom}{\mathrm{Hom}}$This question is a sequel to my ...

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### Cohomology of a differentiable stack: evaluation at a point

I'm reading these Behrend's notes on cohomology of stacks, and I can't get over a detail in the fourth page.
Let $X_\bullet=(X_1\rightrightarrows X_0)$ be a Lie groupoid and let $\mathcal{N}$ be its ...

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1
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### Conservative cocompletion of categories of geometric shapes for homotopy theory

The recent paper
Calin Tataru, Partial orders are the free conservative cocompletion of total orders. arXiv:2404.12924
has shown that the conservative cocompletion of the simplex category $\Delta$ ...

6
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1
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261
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### Fibrations in a model structure for homotopy $n$-types of simplicial sets

Consider the model structure on simplicial sets where the cofibrations are given by monomorphisms and the weak equivalences are given by $n$-equivalences (that is, maps $f \colon X \to Y$ that induce ...

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### What are the centre and trace of the simplex category?

Definition. The centre of a category $\mathcal{C}$ is the set $\mathrm{Z}(\mathcal{C})$ defined by
\begin{align*}
\mathrm{Z}(\mathcal{C}) &\mathbin{\overset{\mathrm{def}}{=}} \int_{A\in\mathcal{C}}...

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0
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### Morphisms in cube category $\Box$ = Compositions of morphisms in simplex category $\Delta$?

Let $\Delta$ be the simplex category. For $m \leq c \leq n$, let $[m] \to [c] \to [n]$ be the composition of two injective morphisms in $\Delta$.
We now define a category $\Box$ with same objects as $\...

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### Isbell duality for simplicial sets

$\newcommand{\IsbellSpec}{\mathsf{Spec}}\newcommand{\IsbellO}{\mathsf{O}}\newcommand{\Nat}{\mathrm{Nat}}$Isbell duality sets up an adjunction (see here for a short abstract summary)
$$\mathsf{O}\dashv\...

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### Whitehead lemma for simplicial Lie algebras

Let $R \to R'$ be a morphism of connected free simplicial Lie algebras. Then the analog of the Whitehead lemma states that if $\pi_* f_{\mathrm{ab}}$ is an isomorphism then $\pi_* f$ is an isomorphism....

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### On Lemma 5.5.16 of Cisinski's "Higher Categories and Homotopical Algebra"

I have a question regarding Section 5 of Cisinski's
"Higher Categories and Homotopical Algebra".
Let us write $\mathbf{sSet}$ and $\mathbf{bisSet}$ for the categories of
simplicial sets and ...

7
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1
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### Bisimplicial spaces as a coequalizer of maps between "simpler" bisimplicial spaces

From a bisimplicial space $T$, one can consider the simplicial spaces $p \mapsto T_{pp}$, $p \mapsto | q \mapsto T_{pq}|$, and $q \mapsto |p \mapsto T_{pq}|$, where $| \cdot|$ denotes geometric ...

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### Multiplicative structure on Čech–Alexander complexes

I have the following basic question on Čech–Alexander complexes.
Let $R$ be a ring and $A$ be an $R$-algebra. To this datum one can attach a cosimplicial ring which assigns to an object $[n]=\{0,1,\...

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0
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### classifications of all weak factorisation systems on a category [duplicate]

Is there an example of a category where all the weak factorisation systems have been classified ? Is this something that people tried to classify ?
This can be done trivially for Sets (see the ...

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### Plus construction on Simplicial Sets?

I had asked this question in Math StackExchange a few days ago, but didn't get any answers. I believe its more suitable to be asked here.
Write $\mathsf{sSet}$ for the category of simplicial sets and $...

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### Is the category of simplicial $R$-modules closed monoidal?

I am trying to understand if the simplicial mapping space for simplicial $R$-Modules (or at least simplicial vector spaces) is adjoint to the (level-wise) tensor product. For simplicity, let me state ...

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### cocycle datum for principal $G$-bundle over base space Delta set

Let $X$ be topological realization of a (finite)
Delta set, $G$ a finite group and $p: P \to X$ a
principal $G$-bundle.
Let's recall the standard fact that more generally any
numerable principal G-...

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0
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### Piecewise construction of a functor from an $(\infty,1)$-category with an orthogonal factorization system

For the simpler case of 1-categories, consider a 1-category $C$ and an orthogonal factorization system $(L,R)$ on $C$. Let $C_L$ and $C_R$ denote the wide subcategories of $C$ corresponding to the ...

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### How to prove a 1-localization of a 1-category is already an $(\infty,1)$-localization?

I don't even know this fits in here or in Mathematics Stack Exchange, but let me ask. I'm new to simplicial stuff, so a good reference would be quite helpful.
Let's say $C$ is a certain category, and ...

2
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1
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### combinatorical description of classifying map for principal $G$-bundle over Delta set

Let $X$ be topological realization of a (finite) Delta set, $G$ a finite group and $p: P \to X$ a principal $G$-bundle. It's standard that the isom' classes of such principal $G$-bundles are ...

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### Are Euclidean spaces $\Delta$-generated?

From the definition of $\Delta$-generated it seems like $\mathbb R$ should be $\Delta$-generated, as $\mathbb R$ is final with respect to all continuous maps $\mathbb R^n \to \mathbb R$.
However, the ...

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0
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### Splitting of $\mathbb{Z}/p\to E\to (\mathbb{Z}/p)^n$ in cohomological terms

Let $d>1$ be an odd integer. Given a simplicial set $X$ and $[\gamma]\in H^2(X,\mathbb{Z}/d)$, there exists a fibration $N\mathbb{Z}/d\to E\to X$, with $$E= X_\gamma:=N\mathbb{Z}/d\times_{\gamma} X....

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### Strøm model structures on the category of simplicial sets

Let $X,Y$ be simplicial sets. A simplicial homotopy is a simplicial map of the form $h:X\times\Delta^1\rightarrow Y$. There are two distinguished maps
$$
in_0:X\cong X\times\Delta^0\xrightarrow{1\...

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### Geometric realization of crossed square

Given a crossed square of groups, you can "totalize" it and get a 2 crossed module in the sense of Conduché "Modules croisés généralisés de longueur 2", then you can apply his ...

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### Explicit examples of 4-cocycles over finite 2-groups

By a (finite) 2-group $X$, I mean a finite group $G$, a finite abelian group $A$, an action of $G$ on $\operatorname{Aut}(A)$, as well as a 3-cocycle $\alpha\in H^3(BG, A)$. They are also equivalent ...

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### G-modules vs. $\Delta(NG)$-modules

Let X be a simplicial set. Its category of simplices, denoted by $\Delta(X)$, is the category whose objects are the pairs $(x,[n])$, with $x\in X_n$, and morphisms $\bar{c}:(y,[m])\to (x,[n])$, where $...

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### Alternative construction for the loop space (?)

There is a way to realize the (infinite) loop space which relies on the (homotopy) totalization of a cosimplicial space. Given a (nice?) topological space $X$, consider the cosimplicial space $X_{\...

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### Why is "everything staying correct" for simplicial spaces?

I recently need a simplicial generalization of some theorem for rigid spaces, namely Theorem A holds for a rigid space $X$ and I want a Theorem $A_\bullet$ for a simplicial rigid space $X_\bullet$. ...

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### Uniqueness of geometric simplicial structure on $\mathbb{R}^n$

By work of Stallings we know that $\mathbb{R}^n$ (for $n\neq4$) has unique simplicial structure up to equivalence. If instead of a general simplicial structure we consider only geometric simplicial ...

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### Canonical form of non-decreasing morphisms

There is a simple lemma that I saw in my algebraic topology class at the University a few years ago (with Vallette): for any non-decreasing morphism $\varphi: [n] \to [m]$ in the category $\Delta$, ...

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1
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### Reference request-Natural equivalence detected pointwise for complete Segal spaces

I am looking for a reference for the following elementary assertion on complete Segal spaces:
Let $A$ be a bisimplicial set and let $W$ be a complete Segal space. A morphism of $W^A$ is an ...

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### What is the intuitive difference between these two simplicial subdivision functors?

$\newcommand{\sset}{\mathsf{sSet}}\newcommand{\poset}{\mathsf{Poset}}\newcommand{\p}{\mathscr{P}^{\mathsf{nd}}}\newcommand{\N}{\mathcal{N}}\newcommand{\sd}{\operatorname{sd}}$Following this paper, I ...

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### Simplicial resolution for commutative group scheme

Let $X$ be a quasi-projective $k$-variety. In this case the symmetric power $S^d(X)$ is well-defined. If $S^\bullet(X)=\bigsqcup_{n>0}S^d(X)$, where we suppose $S^0(X)=\operatorname{spec}(k)$, then ...

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### Spectral sequence for a truncated semi cosimplicial space

Consider the simplicial indexing category $\Delta$. Now, let's denote the subcategory consisting of injections as $\Delta_{inj}$. When we're dealing with a cosimplicial space, which is essentially a ...

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### Hammock localization and free adjoints

The Hammock localization $L^H \mathcal{C}$ of a relative category $(\mathcal{C},\mathcal
{W})$ is a simplicial category defined by Dwyer and Kan as a way to compute the $\infty$-categorical ...

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### Higher categories using just simplicial sets

Is there a definition of $(\infty, n)$-category using just simplicial sets?
This is the case for $n \leq 2$.
Is the forgetful functor from saturated $n$-trivial complicial sets to simplicial sets an ...

3
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1
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### A fiber-like method to show equivalence of infinity categories

Suppose I have a functor of quasi-categories $f: \mathcal{C} \to \mathcal{D}$. I want to show a criterion like: "$f$ is an equivalence of $\infty$-categories if the homotopy fiber of $f$ ...