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.

Filter by
Sorted by
Tagged with
2
votes
0answers
174 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 ...
6
votes
2answers
273 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 ...
0
votes
0answers
69 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 @>&...
7
votes
1answer
387 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 ...
1
vote
0answers
186 views

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 ...
12
votes
3answers
427 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 ...
3
votes
0answers
90 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 ...
8
votes
0answers
136 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 ...
5
votes
0answers
82 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 ...
0
votes
1answer
81 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 $s:X\times X \rightarrow \mathbb{R}$ ...
4
votes
1answer
75 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 ...
1
vote
0answers
49 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 $\...
0
votes
0answers
33 views

The category of connected ribbon graph and its connected component

Let $RG$ be a category of connected ribbon graph, the morphisms are admissible epimorphism or finite composition of contraction. By a ribbon graph we mean a connected graph $\Gamma$ with fixed cyclic ...
2
votes
1answer
67 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 $k$-...
7
votes
0answers
446 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 ...
9
votes
0answers
218 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^\...
2
votes
1answer
146 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 ...
5
votes
1answer
131 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 ...
2
votes
0answers
109 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 $B_\bullet$-...
2
votes
0answers
89 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 ...
2
votes
0answers
64 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/...
2
votes
0answers
59 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....
10
votes
1answer
426 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 ...
9
votes
2answers
261 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 ...
3
votes
0answers
241 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 ...
6
votes
0answers
172 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 ...
6
votes
1answer
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_{(\...
3
votes
0answers
91 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 $...
0
votes
0answers
20 views

Constraints on simplices in hyper sphere packings

Question: what is known about constraints on the side lengths of $n$-simplices that are defined by the centers of $n+1$ kissing $(n-1)$-spheres? The constraints on the curvatures $c_i$ of Soddy ...
2
votes
0answers
85 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 $n$-...
8
votes
1answer
414 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 ...
4
votes
0answers
106 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 \...
5
votes
1answer
211 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} \...
3
votes
1answer
143 views

Derived Category of strictly simplicial algebraic space vs. systems of objects in the derived categories

Let $X_{\bullet}^+$ be a strictly simplicial algebraic space and for a morphism $\delta:[m]\to[n]$ in $\Delta^+$, let $\delta:X_n\to X_m$ also denote the associated map (by abuse of notation). Then ...
2
votes
1answer
249 views

question about notation in HTT of J.Lurie

In page 27 in HTT of J.Lurie, the expression $$\text{Map}_S(X,Y):=Y^X\times_{S^X}\{\phi\}\in \text{Set}_\Delta$$ appears for simplicial set $X,Y,S$ in Warning 1.2.2.2. However, I couldn't understand ...
6
votes
1answer
288 views

What are some "good" examples of Kan simplicial manifolds?

According to the definition 1.1 of the paper Kan Replacement of simplicial manifolds by Chenchang Zhu https://arxiv.org/pdf/0812.4150.pdf, A Kan simplicial manifold is a simplicial manifold $X$ such ...
5
votes
2answers
281 views

Regular or h-regular CW-complex structure for the Poincaré homology sphere

I am looking for a regular (the characteristic maps of the cells are homeomorphisms) or h-regular (the characteristic maps of the cells are homotopy equivalences) CW-complex structure for the Poincaré ...
12
votes
1answer
235 views

Can one show corbordism invariance of the Crane-Yetter state-sum using simplicial methods / are there 'Pachner-like' moves for cobordisms?

Let $\mathcal{C}$ denote some Unitary Braided Modular Fusion Category. It is well known that the Crane-Yetter state-sum, $Z_{CY}(\bullet|\mathcal{C})$ is an oriented-cobordism invariant. In other ...
3
votes
1answer
189 views

A cochain complex using degeneracy maps

In constructing singular homology for a topological space, the boundary operator for the singular chain complex is given as an alternating sum of face maps. The degeneracy maps seem to be discarded in ...
5
votes
1answer
101 views

Why is this condition necessary for the existence of a transferred simplicial model structure?

In chapter II of Goerss and Jardine's text on simplicial homotopy theory, they give a general theorem, Theorem 6.8, for transfer of simplicial model structures across a simplicial adjunction. This ...
4
votes
1answer
186 views

Computing homotopy colimit of a space with free $S^1$-action

Context. I am trying to understand the argument in B.4 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799 (on p147). I am still lost. But from Maxime's helpful ...
7
votes
0answers
147 views

Stabilisation of crossed modules?

D. Conduché has introduced a notion of a stable crossed module ("Modules croisés généralisés de longueur 2", JPAA 34 (1984) pp155–178, doi:10.1016/0022-4049(84)90034-3, Def. 3.1). Is there a ...
5
votes
0answers
181 views

Čech nerve $C(U)$ corresponds to $BG$ in same manner as a hypercover $\mathcal{H}(U)$ to

We can via the bar construction canonically associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps ...
3
votes
0answers
110 views

does geometric realization factor through an endofunctor?

Does the functor of geometric realization of a simplicial set as a topological space, factor through an endofunctor of the category of simplicial topological spaces which does something non-trivial (...
2
votes
0answers
111 views

when is the pullback along the shift (decalage) morphism a direct product, in sSets

What is the meaning of the following condition on a morphism in sSets or simplicial topological spaces: a morphism becomes a direct product after the pullback along the shift(decalage) morphism ? ...
7
votes
0answers
175 views

2d TQFTs with values in simplicial sets and Reedy categories

Let $Cob$ be the category such that $Obj(Cob)$ is $\emptyset\sqcup\mathbb{N}$, with $n$ seen as the union of $(n+1)$ circles numbered from $0$ to $n$, morphisms are (homeomorphism classes of ...
2
votes
1answer
88 views

Kan replacement of finite $\mathbb{Q}$-type simplicial set

Sorry if this question is maybe a bit basic, but it is on a rather specialized topic so I think it is more appropriate for MO than SE. Suppose that $X$ is a simplicial set that has finitely many non-...
4
votes
1answer
150 views

Higher homotopy groups of Joyal fibrant replacements of 2-coskeletal simplicial sets

Suppose $X$ is a 2-coskeletal simplicial set (meaning $X^{Δ^k}→X^{∂Δ^k}$ is an isomorphism for all $k≥3$). What is the easiest example of $X$ such that the Joyal fibrant replacement $Y$ of $X$ is not ...
3
votes
0answers
46 views

Homotopy limits of section spaces

Let $\mathcal{U}$ be an open cover of a topological space $B$. As we see, for example in this question, the associated Cech diagram $B_{\mathcal{U}}$ constitutes a simplicial space with the weak ...
5
votes
1answer
294 views

Contractibility of the category of cosimplicial resolutions

Let $\gamma : \mathcal{C} \to \mathcal{M}$ be a functor and define a cosimplicial resoultion of $\gamma$ as a functor $\Gamma: \mathcal{C} \to \mathcal{M}^{\Delta}$ such that $\Gamma C$ is Reedy ...

1
2 3 4 5
14