# 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.

750
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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
votes

1
answer

148
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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 $f$ ...

4
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1
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219
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### 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
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1
answer

528
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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 $...

7
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0
answers

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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" ...

5
votes

1
answer

197
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### 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

84
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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 ...

8
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2
answers

523
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### 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 ...

9
votes

2
answers

345
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### 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 ...

2
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0
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138
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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 ...

4
votes

1
answer

126
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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 ...

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0
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### 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 ...

3
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1
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93
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### 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 $\mathcal{C}$ ...

3
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1
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### 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}_\...

4
votes

0
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93
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### 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 ...

2
votes

1
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216
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### 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 ...

2
votes

0
answers

116
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### 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 ...

6
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1
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387
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### 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 ...

3
votes

1
answer

98
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### 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 $n$...

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### 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)...

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0
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### 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 ...

6
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1
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### 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}}\...

6
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0
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333
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### 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, ...

4
votes

1
answer

254
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### 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 ...

3
votes

1
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148
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### 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 $n+1$-...

4
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0
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### 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 ...

1
vote

1
answer

178
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### 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 ...

2
votes

1
answer

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### Do there exist smaller simplicial models of barycentric subdivisions?

Let $S$ be a simplicial complex and let $Bary(S)$ denote its barycentric subdivision.
Of course, the geometric realizations of $S$ and $Bary(S)$ are homeomorphic.
However, one issue that arises in ...

9
votes

4
answers

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### Applications of the Dold-Kan correspondence

The Dold-Kan correspondence says essentially that simplicial abelian groups and nonnegative chain complexes of abelian groups are equivalent objects. While this is a very natural statement, I am not ...

4
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0
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99
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### Interplay beween simplicial and Weyl algebra identities

Recall that the (first) Weyl algebra is the algebra generated by $x,y$ with the relation $xy-yx=1$. It can be realized as the algebra of differential operators on $k[x]$, where one generator acts as ...

6
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1
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### Why, if the geometric realisation of a simplicial map $p$ is a (topological) covering map, must $p$ be a (simplicial) covering map?

I essentially am asking for an explanation of the comment under this post by Tom Goodwillie.
In the "Kerodon", Lurie defines a simplicial covering map as follows:
A map $p:E\to X$ of ...

0
votes

0
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186
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### Double complex of simplicial resolution

In his
lectures on condensed mathematics on page 30 Peter Scholze speaks of the double complex of a simplicial resolution. How is this defined?
In the next line, he writes that if $A_\bullet$ is a ...

6
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0
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215
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### Higher Algebra, Propositions 2.3.4.5 and 2.3.4.9

I am reading the proof of Propositions 2.3.4.5 of Higher Algebra by Jacob Lurie. There is a part that I don't understand, and I need someone's help.
In the book, Lurie introduces the notion of ...

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1
answer

632
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### Is there a higher analog of "category with all same side inverses is a groupoid"?

There is a (most likely folklore) theorem - if in a category every morphism has a right inverse then that category is a groupoid. The proof is an honest oneliner: for $x:A\to B$ find $x':B\to A$ with $...

9
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0
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### Applications of the simplex $2$-category and its higher dimensional cousins

The simplex category $\Delta$ has a $2$-category refinement $\Delta_2$ given by the full sub-$2$-category of the $2$-category $\mathsf{Cat}$ spanned by the ordinal categories $𝟘$, $𝟙$, $𝟚:=𝟙\star𝟘...

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0
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### Higher Algebra, Theorem 2.4.3.18 and Remark 2.4.3.6

In his book Higher Algebra, Lurie introduces the notion of generalized $\infty$-operads ($\S$2.3.2). Roughly speaking, a generalized $\infty$-operad is a "family" of $\infty$-operads ...

2
votes

1
answer

138
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### Does the forgetful functor from an over-$(\infty,1)$-category create weakly contractible limits?

Let's say that a limit diagram $\bar p:K^\triangleleft\to\def\sC{\mathscr{C}}\sC$ is a weakly contractible limit if the simplicial set $K$ is weakly contractible (in that $K\to*$ is a weak homotopy ...

6
votes

1
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296
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### Do the various notions of morphism spaces of simplicial sets agree on the underived level?

$\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}$There are (at least) seven kinds of morphism spaces for a simplicial set $X$:
The left-pinched morphism space $\Hom^L_X(x,y)$,
The right-...

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0
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### Simplicial sets and oriented simplicial complexes

$\DeclareMathOperator\Sing{Sing}$I'm writing a paper about simplicial sets and how they may “replace” simplicial complexes in some known results. To do this, we need to check that they induces the ...

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0
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### Functorial identification of the mapping spaces of the arrow category of an $\infty$-category

Let $\mathcal{C}$ be a small $\infty$-category. Using the straightening-unstraightening construction, we can define a hom-functor of $\mathcal{C}$ as a functor $\mathcal{C}(-,-):\mathcal{C}^{\mathrm{...

6
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1
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227
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### On morphism spaces of $(\infty,2)$-categories failing to be $\infty$-categories

As noted in Tag 01WZ of Kerodon, the morphism spaces of an $(\infty,2)$-category $C$ can fail to be $\infty$-categories, in contrast to the pinched morphism spaces of $C$.
This feels very ...

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0
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220
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### Homotopy theory of cospaces (or $\infty$-cogroupoids)

Is there a good homotopy theory for cospaces, where a cospace (or $\infty$-cogroupoid) would be a cosimplicial set satisfying some appropriate dual version of the Kan condition?
One point I'm curious ...

3
votes

1
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145
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### A projective-cofibrant replacement in $\mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ such that $\operatorname{ev}_0Q(S) \cong \operatorname{ev}_0S$

Consider the category of simplicial presheaves $\mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ endowed with the projective model structure, i.e. weak equivalences and fibrations are point-wise.
I need a ...

3
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2
answers

299
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### Pushout along weak equivalence gives weakly equivalent object

This question arose through reading "Interactions between homotopy theory and algebra" (the first chapter by Goerss and Schemmerhorn). In particular, I am struggling with the proof of ...

6
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3
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459
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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 ...

0
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1
answer

181
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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 ?
...

0
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0
answers

85
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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 $...

2
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1
answer

68
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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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0
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57
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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 ...

4
votes

1
answer

140
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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 ...