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
questions
6
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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 ...
- 763
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 $f$ ...
- 1,772
4
votes
1
answer
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 ...
- 1,417
9
votes
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 $...
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votes
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" ...
- 703
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. ...
- 1,440
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 ...
- 1,440
8
votes
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 ...
- 1,440
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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 ...
- 1,217
2
votes
0
answers
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 ...
- 5,272
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 ...
- 1,772
0
votes
0
answers
46
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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 ...
- 5,627
3
votes
1
answer
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 $\mathcal{C}$ ...
- 486
3
votes
1
answer
153
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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}_\...
- 19.3k
4
votes
0
answers
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 ...
- 1,063
2
votes
1
answer
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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 ...
- 503
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 ...
- 21
6
votes
1
answer
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 ...
- 503
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 $n$...
- 1,872
6
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0
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114
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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
$$
\underline{\mathrm{Hom}}(K,L)...
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1
vote
0
answers
60
views
Powersets of simplicial sets vs. powersets of topological spaces
Motivation. Recently I've been trying to understand how well- or ill-behaved are the many different powerset topologies one can put on the powerset of a topological space, and in particular I'm trying ...
- 9,575
6
votes
1
answer
206
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Homotopical properties of powersets of simplicial sets
Given a simplicial set $X_\bullet$, define its powerset simplicial set $\mathcal{P}_\bullet(X)$ as the composition
$$\Delta^\mathsf{op}\xrightarrow{X_\bullet}\mathsf{Set}\xrightarrow{\mathcal{P}}\...
- 9,575
6
votes
0
answers
333
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, ...
4
votes
1
answer
254
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 ...
- 1,440
3
votes
1
answer
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 $n+1$-...
- 503
4
votes
0
answers
101
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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 ...
- 233
1
vote
1
answer
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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 ...
- 503
2
votes
1
answer
121
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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
1k
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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 ...
- 99
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votes
0
answers
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 ...
- 5,627
6
votes
1
answer
313
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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 ...
- 333
0
votes
0
answers
186
views
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 ...
- 1,662
6
votes
0
answers
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 ...
- 1,440
15
votes
1
answer
632
views
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 $...
- 17.1k
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answers
217
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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...
- 9,575
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0
answers
495
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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 ...
- 1,440
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 ...
- 587
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votes
1
answer
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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-...
- 9,575
1
vote
0
answers
119
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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 ...
- 11
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0
answers
202
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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{...
- 1,440
6
votes
1
answer
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 ...
- 9,575
4
votes
0
answers
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 ...
- 9,575
3
votes
1
answer
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 ...
- 33
3
votes
2
answers
299
views
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
votes
3
answers
459
views
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 ...
- 1,440
0
votes
1
answer
181
views
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 ?
...
- 317
0
votes
0
answers
85
views
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 $...
- 317
2
votes
1
answer
68
views
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
$\...
- 5,272
1
vote
0
answers
57
views
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 ...
- 317
4
votes
1
answer
140
views
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 ...
- 317