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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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,\...
Stabilo's user avatar
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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 ...
user524793's user avatar
11 votes
1 answer
349 views

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 $...
wind's user avatar
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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 ...
SetR's user avatar
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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-...
JackYo's user avatar
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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 ...
gksato's user avatar
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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 ...
gksato's user avatar
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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 ...
JackYo's user avatar
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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 ...
William B.'s user avatar
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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....
Antoine's user avatar
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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\...
Tyrone's user avatar
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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 ...
clovis chabertier's user avatar
3 votes
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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 ...
Andi Bauer's user avatar
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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 $...
Antoine's user avatar
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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_{\...
Andrea Marino's user avatar
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1 answer
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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 $X_\bullet$. ...
Johnny's user avatar
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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 ...
Prateep's user avatar
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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 $\Delta$, ...
Andrea Marino's user avatar
2 votes
1 answer
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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 ...
Ken's user avatar
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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 ...
FShrike's user avatar
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3 votes
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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 ...
Sam's user avatar
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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 ...
happymath's user avatar
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1 answer
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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 ...
Simon Henry's user avatar
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6 votes
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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 ...
Daniel Bruegmann's user avatar
3 votes
1 answer
178 views

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 $f$ ...
Andrea Marino's user avatar
4 votes
1 answer
240 views

Can one bypass the geometric realization in the definition of algebraic $K$-theory?

I believe there is no good notion of homotopy groups for an arbitrary simplicial set $S$. However, when $S$ is fibrant - meaning that $S\to *$ is a fibration - there is a definition. The singular ...
Stabilo's user avatar
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What is known about the homotopy type of the classifier of subobjects of simplicial sets?

For the presheaf topos $\mathrm{PSh}(C)$, the subobject classifier is the presheaf $\Omega$ such that For $c \in C$, $\Omega(c)$ is the set of all subobjects of the functor $\mathrm{Hom}(-, c)$ For $...
Arshak Aivazian's user avatar
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Can cyclic and simplicial objects be related in a similar way to how the species of linear orders is the derivative of the species of cyclic orders?

The derivative of a combinatorial species $S: core(FinSet) \to core(FinSet)$ is given by $S^\prime [N] = S[N\sqcup 1]$. Intuitively, an $S^\prime$-structure is built by introducing a "hole" ...
Mathemologist's user avatar
5 votes
1 answer
219 views

Cofinal maps from posets (HTT, 4.2.3.16)

I do not understand the proof of Variant 4.2.3.16 of Higher Topos Theory by Jacob Lurie, and I need help. Variant 4.2.3.16 asserts the following: ($\diamond$) Let $K$ be a finite simplicial set. ...
Ken's user avatar
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4 votes
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Localization and space of morphisms

I have a question regarding the proof of Proposition 2.19 of Factorization homology of topological manifolds by Ayala and Francis. In the final paragraph of the proof (more specifically, in the second ...
Ken's user avatar
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8 votes
2 answers
564 views

Homotopic but not equivariantly homotopic maps

Let $G$ be a topological (or simplicial) group, let $X$ and $Y$ be $G$-spaces, and let $f,f':X\to Y$ be $G$-maps which are homotopic as maps of spaces. In general, $f$ and $f'$ may (of course) fail to ...
Ken's user avatar
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9 votes
2 answers
379 views

Simplicial sets with horn filling conditions up to some fixed degree

Let $X_\bullet$ be a simplicial set such that some horn filling condition (inner horns fill/inner horns fill uniquely/all horns fill) holds up to dimension $n$ (i.e. for $\Lambda_i[p]$ for all $p\leq ...
Tim's user avatar
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2 votes
0 answers
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Motivation for working with augmented objects in homological or higher algebra

I would like to understand if there is deeper reason/motivation behind augmentations in homological algebra. Recall classically in homology if there is a complex of free $R$-modules ($R commutative ...
user267839's user avatar
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4 votes
1 answer
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Homotopy totalization and chains - reference

Simple case. Take $X_{\bullet}$ a cosimplicial space. Is it true that the chain complex of its homotopy totalization is quasi-isomorphic to the homotopy totalization of its chain complex? Because of ...
Andrea Marino's user avatar
0 votes
0 answers
49 views

Finitely continuous fibrant replacement functor for localization of simplicial presheaves with projective model structure

Let $C$ be a model category given by generators and relations in the sense of Dugger (that is, $C$ is a left Bousfield localization of a global projective model model structure on simplicial ...
Arshak Aivazian's user avatar
3 votes
1 answer
104 views

Injective model structure for simplicial presheaves

I am reading the paper by Jardine and Goerss, Localization theories for simplicial presheaves and having troubles with understand an argument. In this paper, the two authors considered $\mathcal{C}$ ...
Alexey Do's user avatar
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3 votes
1 answer
187 views

Monoidal structure on simplical model category of chain complexes

For $k$ a field (the case I am interested in, but the question makes sense over any dga), $\mathrm{Ch}_\bullet(k)$ its projective model category of unbounded chain complexes (here), $\mathrm{sCh}_\...
Urs Schreiber's user avatar
4 votes
0 answers
96 views

Convolution algebra of a simplicial set

Consider a simplicial set $X^\bullet$ with face maps $d_i$ (assume the set is finite in each degree so there are no measure issues). Then given two functions $f,g:X^1\to \mathbb{C}$ one can form their ...
Josh Lackman's user avatar
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2 votes
1 answer
226 views

Viewing simplicially the Stone space of types of a first-order theory

Let $T$ be a first-order theory, let $M$ be a monster model of $T$. For a set $B\subset M$, let $S_n(B) := M^n/\operatorname{Aut}^T(M)$ be the Stone space of complete $n$-types of $T$ with parameters ...
user494312's user avatar
2 votes
0 answers
118 views

Multiplicativity of $\operatorname{Tot}_n$

Suppose $R^\bullet$ is a cosimplicial ring. I think this induces a ring structure on $\operatorname{Tot}(R^\bullet)$-- I couldn't find a classical reference, but I did find a recent paper of Batanin ...
user502020's user avatar
5 votes
1 answer
503 views

Is the singular simplicial complex functor $\operatorname{Sing}_\bullet:\operatorname{Top} \to \operatorname{sSets}$ fully faithful for nice spaces?

$\DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\Top}{Top} \DeclareMathOperator\sSets{sSets}$Is there a category of “sufficiently nice” topological spaces such ...
user494312's user avatar
3 votes
1 answer
150 views

Simplicial set from all orderings of simplicial complex

Given an abstract simplicial complex $K$ on a set of vertices $V$, we can form a semi-simplicial set by $F(K)$ sending $F(K)_n$ to be the set of ordered $(n+1)$-tuples of vertices in $V$ forming an $n$...
xir's user avatar
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6 votes
0 answers
127 views

Mapping space between $n$-groupoids is an $n$-groupoid

Consider two simplicial sets $K$ and $L$. Their mapping space (or mapping complex) is the internal hom of simplicial sets, i.e. $\underline{\mathrm{Hom}}(K,L)$, where $$ \underline{\mathrm{Hom}}(K,L)...
SetR's user avatar
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1 vote
0 answers
71 views

Powersets of simplicial sets vs. powersets of topological spaces

Motivation. Recently I've been trying to understand how well- or ill-behaved are the many different powerset topologies one can put on the powerset of a topological space, and in particular I'm trying ...
Emily's user avatar
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6 votes
1 answer
208 views

Homotopical properties of powersets of simplicial sets

Given a simplicial set $X_\bullet$, define its powerset simplicial set $\mathcal{P}_\bullet(X)$ as the composition $$\Delta^\mathsf{op}\xrightarrow{X_\bullet}\mathsf{Set}\xrightarrow{\mathcal{P}}\...
Emily's user avatar
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6 votes
0 answers
369 views

Higher Algebra, Section 2.2.2

I am reading Section 2.2.2 of Higher Algebra by Jacob Lurie. There are proofs which I cannot understand, so I need someone's help. First, I cannot understand the proof of Lemma 2.2.7. In the proof, ...
Yutaro Mikami's user avatar
4 votes
1 answer
267 views

HTT, Remark 4.2.4.5

In his book Higher Topos Theory, Lurie proves that in some favorable cases, functor categories of $\infty$-categories admits a rigid model. More precisely, he proves in Proposition 4.2.4.4 the ...
Ken's user avatar
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3 votes
1 answer
182 views

Defining the classifying space of a group acting on a set

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gal{Gal}$Recall that $n$-simplices of the classifying space (simplicial set) $BG$ of a group are orbits of the diagonal action by shifts on $n+1$-...
user494312's user avatar
4 votes
0 answers
105 views

Does integration induce a Kan fibration between the mapping spaces of CDGAs and cochain complexes over $\mathbb{Q}$?

For two CDGAs $A$ and $B$ over $\mathbb{Q}$, the mapping space $\text{Map}_{CDGA}(A,B)$ is the simplicial set with $n$-simplices $$ \Phi : A \to B \otimes \Omega^*(\Delta^n) $$ and simplices maps ...
kelly maggs's user avatar
1 vote
1 answer
210 views

The simplicial set with a unique non-degenerate simplex in each dimension

There is a unique simplicial set with a unique non-degenerate simplex in each dimension, (updated) and such that all faces of the non-degenerate simplex are non-degenerate. Does it have a name, and ...
user494312's user avatar

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