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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8 votes
0 answers
115 views

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 votes
1 answer
87 views

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 ...
3 votes
0 answers
139 views

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,\...
0 votes
0 answers
17 views

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 ...
8 votes
1 answer
154 views

Morphisms of hammocks in the simplicial localization

Let $\mathcal{C}$ be a category together with a wide subcategory $\mathcal{W} \subset \mathcal{C}$. In Calculating Simplicial Localizations by Dwyer and Kan, a morphism of hammocks is defined to ...
7 votes
1 answer
136 views

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 ...
11 votes
1 answer
617 views

Is the Thomason model structure on Cat simplicial? Is it a monoidal model category?

The Thomason model structure on the category of small categories is transferred from the Quillen model structure on simplicial sets along the right adjoint $Ex^2 \circ N$ (where $N$ is the nerve), i.e....
11 votes
1 answer
371 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 $...
2 votes
1 answer
171 views

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 ...
2 votes
0 answers
62 views

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-...
2 votes
1 answer
174 views

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 ...
1 vote
0 answers
127 views

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 ...
5 votes
1 answer
255 views

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 ...
2 votes
0 answers
79 views

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....
14 votes
3 answers
583 views

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\...
2 votes
0 answers
166 views

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 ...
3 votes
0 answers
75 views

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 ...
2 votes
0 answers
73 views

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 $...
2 votes
0 answers
92 views

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_{\...
1 vote
0 answers
138 views

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 ...
1 vote
1 answer
361 views

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 $X_\bullet$. ...
2 votes
1 answer
80 views

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 ...
0 votes
0 answers
28 views

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 $\Delta$, ...
5 votes
2 answers
511 views

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 ...
1 vote
0 answers
99 views

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 ...
3 votes
0 answers
158 views

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 ...
8 votes
2 answers
804 views

Reference for homotopy colimit = total complex

I'm looking for a reference for the following fact: take a simplicial chain complex $ X:\Delta^{op}\to Ch_{\geq 0}(\mathcal A)$ for $\mathcal A$ a nice abelian category (say, cocomplete with enough ...
5 votes
1 answer
215 views

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 ...
6 votes
0 answers
204 views

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 ...
9 votes
2 answers
383 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 ...
3 votes
1 answer
268 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 ...
3 votes
1 answer
179 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 $f$ ...
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 ...
9 votes
1 answer
553 views

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 $...
2 votes
1 answer
334 views

Hypervolume under the square of an n-simplex

I posted this question at Mathematics SE, reformulated and posted again both times without much luck. I also asked a math professor at my university who suggested I post it here. Hopefully, it is ...
7 votes
0 answers
80 views

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" ...
5 votes
1 answer
220 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. ...
4 votes
0 answers
95 views

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 ...
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 ...
2 votes
0 answers
144 views

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 ...
4 votes
1 answer
169 views

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 ...
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 ...
24 votes
6 answers
2k views

Simplicial model of Hopf map?

The Hopf fibration is a famous map $S^3\to S^2$ with fiber $S^1$, which is the generator in $\pi_3(S^2)$. We can model this map in terms simplicial sets by taking the singular simplicial sets of these ...
9 votes
2 answers
892 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 ...
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 $\mathcal{C}$ ...
4 votes
1 answer
397 views

Contiguity for simplicial maps between simplicial sets

I begin by recalling the definition of contiguous simplicial maps between abstract simplicial complexes: Definition. Two simplicial maps $\varphi,\psi\colon K \to L$ are said to be contiguous if for ...
3 votes
1 answer
190 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}_\...
5 votes
1 answer
606 views

Why is the straightening functor the analogue of the Grothendieck construction?

In classical category theory, there is the notion of functor (co)fibered in groupoids. Furthermore, via Grothendieck construction we have an equivalence between pseudo functors into the category of ...
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 ...
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 ...

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