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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Kan–Thurston theorem and R-completion

A corollary of the Kan and Thurston theorem states that the space $X$ (path connected) can be recovered, up to homotopy, by applying the fiber-wise $\mathbb{Z}$-completion functor of Bousfield–Kan (...
Misha Kornev's user avatar
2 votes
0 answers
141 views

Chain-level representability of simplicial cohomology

There's a simple construction for a classifying space $K(G,1)$ of a finite group $G$ as an infinite simplicial complex consisting of one $i$-simplex for each possible simplicial $1$-cocycle restricted ...
Andi Bauer's user avatar
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When is the fat join a monoidal structure?

This question is about the following general construction. Definition: Let $(\mathcal C, \otimes)$ be a cocomplete, monoidal biclosed category whose unit $\ast$ is terminal. Let $I$ be an "...
Tim Campion's user avatar
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22 votes
2 answers
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Why are quasi-categories better than simplicial categories?

There are many models for $(\infty,1)$-categories: simplicial categories, Segal categories, complete Segal spaces, and quasi-categories. Doubtlessly the model most used to do higher category theory in ...
user475383's user avatar
7 votes
1 answer
326 views

How do the various homotopy 2-categories compare?

There are various models of $\infty$-categories floating around, so there are as many models of the associated homotopy 1- and 2-categories. Because the relations between the former are worked out in ...
Jonas Linssen's user avatar
2 votes
0 answers
215 views

Formally étale maps of animated $k$-algebras

In Lurie's DAG, he defines what it means for a natural transformation $T:\mathcal{F}\to\mathcal{F}'$ of functors $\mathcal{F},\mathcal{F}':\mathcal{SCR}\to\mathcal{S}$ to be formally étale. Namely, it ...
Eric's user avatar
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Is there any research about secant varieties by using homotopical algebra or simplicial methods?

Let $A$ be a finitely generated $\mathbb{C}$-algebra with $proj A$ is a projective variety $X \subset \mathbb{P}^n$. The join $J(X,X)$ can be represented by $proj (A\otimes_kA)$ and also we can ...
Doyoung Choi's user avatar
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Reference request - Two notions of suspension /loop space agree for simplicial objects

In Goerss-Jardine they point out that there are two reasonable definitions of the (reduced) suspension of a simplicial set. One is the smash product with $S^1$ and one is the join with $S^0$, the &...
Patrick Nicodemus's user avatar
5 votes
1 answer
239 views

Example of a non-$\infty$-category whose homotopy category is a groupoid

What is an example of a simplicial set $S$ such that its homotopy category $hS$ is a groupoid, but such that $S$ is not an $\infty$-category? I know that if $S$ is an $\infty$-category, then $S$ is a ...
user997814's user avatar
4 votes
0 answers
98 views

Simplicial spaces and reflexive coequalisers

Let $X_\bullet$ be a simplicial space. Consider the reflexive coequaliser of $X_1\rightrightarrows X_0$, which we call $X$. Then we clearly have a map $\varphi\colon |X_\bullet|\to X$, where $|X_\...
FKranhold's user avatar
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The localization of the span category

Suppose one has a model category $C$ with its class of weak equivalences $W$. It is possible to form a separate homotopical category $(C_{\operatorname{span}},W_{\operatorname{span}})$ which has ...
Connor Malin's user avatar
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2 votes
1 answer
162 views

Explicitly calculating the homotopy fiber of a 3-cocycle in the category of simplicial sets

I am stuck at a critical step in my master's thesis. If someome can help me out here, I will give appropriate credit. We know that the data of a 2-group $G$ can be given by a group $\tau_0(G)$ and a 3-...
Alexander Praehauser's user avatar
11 votes
2 answers
1k views

Algebraic topology and homotopy theory with simplicial sets instead of topological spaces

To quote Kerodon: In fact, it is possible to develop the theory of algebraic topology in entirely combinatorial terms, using simplicial sets as surrogates for topological spaces. A similar quote can ...
user469290's user avatar
4 votes
1 answer
69 views

Simplicial polytope with regular cones

Let $P$ be a convex simplicial polytope in $\mathbb{R}^n$. Can we find a convex simplicial polytope $P_0$ in $\mathbb{R}^n$ combinatorially equivalent to $P$, satisfying the following condition: The ...
user73577's user avatar
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Geometric realisation of smooth $\infty$-stacks

Let $Sh^\infty(\mathsf{Man})$ denote the $\infty$-category of sheaves of $\infty$-groupoids over the site $\mathsf{Man}$ of smooth manifolds (if you prefer, that's the model category of simplicial ...
André Henriques's user avatar
4 votes
1 answer
224 views

Kan Complexes, proof of extension of a map to a product

I'm reading a book about Kan Complexes in Simplicial Homotopy Theory (Curtis), and came across this theorem at the beginning : if $f : \Delta[n] \times \Lambda^k[m] \to K$ is a simplicial map and $K$ ...
Raphaël Kalfon's user avatar
1 vote
1 answer
331 views

Proving that the simplex category is generated by the face and generacy maps

Note: This is not intended to be a research level question, but concerns graduate level material. Theorem. The opposite $\Delta^\mathrm{op}$ of the simplex category $\Delta^\mathrm{op}$ (as usually ...
user984603's user avatar
3 votes
1 answer
280 views

Can homotopy limits of simplicial sheaves be calculated (correctly) using sheaves of Kan complexes?

$\DeclareMathOperator\holim{holim}$ Let $sSh$ be the category of simplicial sheaves on some site (I like using the psychological crutch of the site having enough points; further, to clarify a bit, &...
rvk's user avatar
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0 answers
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Why is this class of right-anodyne maps closed under pushouts?

Let $S$ be the class of all right-anodyne maps $r$ such that the pullback of $r$ along any left fibration is again right-anodyne. According to Land's book on $\infty$-categories (specifically the ...
Doron Grossman-Naples's user avatar
2 votes
0 answers
232 views

Why is $\operatorname{Hom}^R_{\mathcal{C}}(X, Y )$ the fiber of $\mathcal{C}_{\backslash Y} \to \mathcal{C}$ (Lurie's HTT)

Reading Jacob Lurie's Higher Topos Theory I not understand the proof of the "only if" part in Proposition 1.2.12.4. It states Proposition 1.2.12.4. Let $\mathcal{C}$ be an $\infty$-category ...
user267839's user avatar
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6 votes
2 answers
601 views

Simplicial set construction of the classifying space

What would be the best book, article, or otherwise to reference for the specific construction of the classifying space for a discrete group $G$ which goes as follows?: Regard $G$ as a category with ...
Xindaris's user avatar
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0 answers
141 views

A fibration is a map which has the right lifting property with respect to injections that are weak equivalences

As mentioned in Motivic Homotopy Theory, an alternative criterion for $X\rightarrow Y$ to be a fibration is that the diagram $$\require{AMScd} \begin{CD} A @>>> X;\\ @VV{i}V @VVV \\ B @>&...
XT Chen's user avatar
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7 votes
1 answer
533 views

Is $\oplus$ the only monoidal structure on the simplex category?

Simplicial sets are presheaves on the simplex category $\Delta$, while augmented simplicial sets are presheaves on $\Delta_+$, the augmented simplex category. Because Day convolution allows us to ...
Emily's user avatar
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1 vote
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History of simplicial complex

It is easy to find the definition of a simplicial complex: https://en.wikipedia.org/wiki/Simplicial_complex I am interested in the history and first occurrences of the concept. When did people start ...
Till's user avatar
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12 votes
3 answers
504 views

Small simplicial set models for BG

Let $F$ be a finite group. Is there a model for $BF$ as a simplicial set such that the number of nondegenerate $n$-simplices grows at most polynomially? For example the Bar construction has the ...
HenrikRüping's user avatar
4 votes
0 answers
210 views

A notion of "generalized nerve" of categories enriched over a presheaf

Let $\mathcal{C}$ be a small category, $p : \mathcal{C} \to \mathsf{Cat}$ a functor, and $\mathsf{P} = \mathrm{PSh}(\mathcal{C})$ the category of presheaves over $\mathcal{C}$ valued in sets. The ...
Lorenzo Riva's user avatar
8 votes
0 answers
268 views

(Homotopy) inverse limits of towers of spaces or simplicial sets - reference request

Suppose we have a tower of Kan fibrations between Kan complexes: $$ X_0 \xleftarrow{f_0} X_1 \xleftarrow{f_1} X_2 \xleftarrow{f_2} \dotsb $$ From this we get a commutative diagram of topological ...
Neil Strickland's user avatar
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0 answers
100 views

A simplicial analogue of a Hurewicz fibration

Let $f: X \to Y$ be a map of simplicial sets. Then there is an obvious simplicial version of demanding that $f$ be a ``Hurewicz'' fibration. Whether for every simplicial set $S$, $f$ has the lifting ...
GuestUser's user avatar
0 votes
1 answer
229 views

Simplex invariants?

Let $k=s$ be a positive definite symmetric function and a simililarity over the natural numbers such that $k(a,a) = 1 $ for all natural numbers $a$. A similarity $s:X\times X \rightarrow \mathbb{R}$ ...
mathoverflowUser's user avatar
6 votes
1 answer
104 views

Non-enriched Bousfield localizations

We know that whenever we have a Bousfield localization between two simplicial model categories, this gives rise to a reflective subcategory in $\infty$-categories (or coreflective, depending on the ...
Giulio Lo Monaco's user avatar
1 vote
0 answers
58 views

Generalisation of spanning tree in simplex

This is a question I asked on Stackexchange (https://math.stackexchange.com/q/4004734/400240) but I did not receive an answer (I think it was too hard). Per someone's suggestion I put it here. Let $\...
IAnemaet's user avatar
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2 votes
1 answer
96 views

A space $X$ is $k$-truncated iff $\text{Fun}(S,X)$ is $k$-truncated

This is probably a simple question but I can't seem to find any references for it. Call a Kan complex $X$ $k$-truncated if $\pi_n(X, x) = 0$ for all $x \in X_0$ and $n > k$. Claim: $X$ is $k$-...
Emilio Minichiello's user avatar
8 votes
0 answers
847 views

Nonabelian variants of the Breen-Deligne resolution

The Breen-Deligne resolution is an unusual functorial resolution of an abelian group A by finite direct sums of free abelian groups of the form $\Bbb Z[A^n] = Free_{Ab}(A^n)$. It makes several ...
Tyler Lawson's user avatar
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9 votes
0 answers
268 views

A kind of algebraic sphere?

Let $Sch$ be the 1-category of schemes. Is there a cosimplicial scheme $D^\bullet$ and a sequence of schemes $S_1, S_2, ...$ such that the geometric realization of the simplicial set $Hom_{Sch}(D^\...
kiran's user avatar
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2 votes
1 answer
164 views

Does the monoidal structure on semisimplicial sets preserve fibrant objects?

The category of semisimplicial sets has the structure of a monoidal category by the geometric product $\otimes$, see for example Rourke and Sanderson's paper '$\Delta$-sets I: Homotopy Theory'. This ...
EBP's user avatar
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6 votes
1 answer
166 views

Join as a bifunctor

I have been reading these great notes by Charles Rezk, and one thing that has been bothering me is the join construction. To solve lifting problems in quasicategory theory we use the Leibniz ...
Emilio Minichiello's user avatar
2 votes
0 answers
138 views

$\Omega^1_{B_\bullet/A_\bullet}$ is acyclic if $A_\bullet \to B_\bullet$ is quasi-isomorphism

Let $A_\bullet \to B_\bullet$ be a quasi-isomorphism of simplicial rings in the sense of (P.62, I.3.1.7, Complexe Cotangent et Déformations I, Luc Illusie). Then, we define the simplicial $B_\bullet$-...
Kenny Lau's user avatar
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2 votes
0 answers
244 views

Mapping spaces of simplicial model categories and quasicategories

Let $M$ be a simplicial model category, $M^o$ its full subcategory of bifibrant objects. The axioms of a simplicial model category guarantee that $M^o$ is enriched in Kan complexes. Thus the homotopy ...
Emilio Minichiello's user avatar
2 votes
0 answers
76 views

Left anodyne is covariant equivalence

I have a question about the proof of left anodyne between two simplicial sets over $S$, where $S$ is a simplicial set, is covariant equivalence. In the proof (HTT 2.1.4.6 or https://ncatlab.org/nlab/...
Peter Liu's user avatar
  • 253
2 votes
0 answers
63 views

homotopy coherent G-action on tensor product of complexes

Let $G$ be a discrete group and $k$ a field. Suppose $C_1$ and $C_2$ are complexes over $k$ with homotopy coherent actions of $G$ in the sense of Cordier (I've been reading https://arxiv.org/pdf/1801....
mathdonkey's user avatar
12 votes
1 answer
802 views

Modern proofs for simplicial localizations

I know that the references usually regarded as standard for simplicial localizations are the Dwyer and Kan's three articles from the 80's. I would be interested in a more modern approach to the ...
Giulio Lo Monaco's user avatar
9 votes
2 answers
323 views

Simplicial spaces internally to simplicial sets

I am a master’s student with interest in topos theory and its applications (motivated by Ingo Blechschmidt’s thesis, as seems to be usual). After finding out about some of the uses of simplicial ...
César Iglesias's user avatar
3 votes
0 answers
262 views

All functors "are" left adjoints, and applications?

Throughout this thread, let us assume smallness. All functors "are" left adjoints Let $D \xrightarrow{F} C$ be any functor, which induces $$ D \xrightarrow{F} \hat{C}$$ by compositing the ...
Student's user avatar
  • 5,008
6 votes
0 answers
187 views

Base-change for simplicial spaces

Base-change for simplicial spaces Let us say that a map of simplicial spaces $X_* \to Y_*$ is a base-change if for all $n$ the canonical map $$ X_n \to (X_0)^{n+1} \times_{(Y_0)^{n+1}}^h Y_n $$ is ...
Jan Steinebrunner's user avatar
6 votes
1 answer
1k views

Proposition 5.13 (ii) in Scholze's Perfectoid Spaces

In Proposition 5.13 (ii) in Scholze's Perfectoid Spaces, we have $R \to S$ a morphism of $\Bbb F_p$-algebras and the assumption that the relative Frobenius $\Phi_{S/R}$ induces an isomorphism $R_{(\...
Kenny Lau's user avatar
  • 435
3 votes
0 answers
109 views

"Boundaries" in Free Simplicial Monoids

I suspect that this has been addressed somewhere already, but I cannot find anything. Let $\hat{\Delta}$ denote simplicial sets, and $Mon\hat{\Delta}$ simplicial monoids. There is a forgetful functor $...
Mathemologist's user avatar
2 votes
0 answers
111 views

Sufficient coordinate-free condition for points being co-spheric

Question: is there a theorem that guarantees that $\mathcal{P}\subset\mathbb{E}^n$ is finite set of points in a Euclidean space and all radii of the $(n-1)$-spheres that are defined by the $n$-...
Manfred Weis's user avatar
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8 votes
1 answer
550 views

About definition of homotopy colimit of Kan and Bousfield

In Bousfield and Kan's book"Homotopy Limits, Completions and Localizations",they define homotopy direct limit for system of pointed simplicial sets(Ch XII S2 2.1 P327), while they define ...
Allen Lee's user avatar
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4 votes
0 answers
127 views

Example of non reduced representable functor from simplicial rings

Let $\text{sCR}_k$ be the over category of simplicial commuative rings (so objects are maps $R \to k$), where $k$ is the simplicial field generated by a non-derived field of the same name. Let $R \in \...
Martin Ortiz's user avatar
6 votes
1 answer
326 views

Elementary proof of the exactness of Čech complex associated to a hypercovering ("Illusie's Conjecture")

Let $\mathcal{E}$ be a sheaf of abelian groups on a topological space (or a site). For an open covering $\mathfrak{U} = (U_i)_i$, it is well known that the augmented Čech complex $0 \to \mathcal{E} \...
Ingo Blechschmidt's user avatar

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