Questions tagged [simplicial-presheaves]
A simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets).
40
questions
2
votes
1
answer
96
views
Is a left Bousfield localization of simplicial presheaves a locally cartesian closed model category?
Let $\mathcal{C}$ be a small category and let $\mathcal{M} = \operatorname{sPre}(C)$ be the model category of simplicial presheaves on $\mathcal{C}$ with the injective model structure.
Let $S$ be a ...
- 568
0
votes
0
answers
48
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 ...
- 6,006
3
votes
1
answer
100
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 $\mathcal{C}$ ...
- 536
3
votes
0
answers
116
views
Gluing data for $\infty$-sheaves?
Let $\mathcal{F}$ and $\mathcal{G}$ be two $\infty$-sheaves on $X$ resp. $Y$. I want to understand exactly when we can "glue" $\mathcal{F}$ and $\mathcal{G}$ to give a $\infty$-sheaf on $X\...
- 31
3
votes
1
answer
274
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, &...
- 553
1
vote
0
answers
95
views
Invariance of categories of sheaves (on simplicial presheaves) under (local) weak equivalence
Let $\mathcal{C}$ be a Grothendieck site (secretly in my head I am thinking of Hausdorff topological spaces with open covers; if I am daring I might be thinking of the big etale site on complex ...
- 553
6
votes
1
answer
373
views
Homotopy quotients, fixed points and stalks of simplicial (pre)sheaves
$\DeclareMathOperator\holim{holim}\DeclareMathOperator\hocolim{hocolim}$Let $\mathcal{F}$ be a simplicial (pre)sheaf on some site $\mathcal{C}$ (assume the site has enough stalks; if you like also ...
- 553
4
votes
1
answer
178
views
Checking that (hyper) sheafification is fibrant in local projective model structure on simplicial presheaves
Context and Notation
Let $X$ be a manifold and $\mathcal{C} = Op(X)$ be the category of open subsets of $X$ with inclusions. I then consider the projective model structure on the (simplicial model) ...
- 1,331
5
votes
0
answers
99
views
On how Simpson's model structure on Tamsamani $n$-prenerves is cofibrantly generated
I was reading through "A closed model structure for $n$-categories, internal $Hom$, $n$-stacks and generalized Seifert Van-Kampen" by Simpson and was struggling to piece together the ...
- 587
5
votes
0
answers
249
views
Stalk of motivic homotopy sheaves
In contrast to "classical" homotopy theory, in the motivic homotopy theory, we don't have homotopy group but rather homotopy sheaves in the Nisnevich topology, which is associated to the ...
- 4,136
3
votes
1
answer
387
views
Whitehead Theorem in $\mathbb{A}^1$-homotopy theory
I'm reading "UNSTABLE MOTIVIC HOMOTOPY THEORY" by Kirsten Wickelgren and Ben Williams (https://arxiv.org/pdf/1902.08857.pdf). There they have a version of Whitehead's Theorem, namely Prop ...
- 4,136
3
votes
0
answers
124
views
Preserves naively $\mathbb{A}^{1}$-homotopic maps
I've been studying $\mathbb{A}^{1}$-homotopy recently and would like some guidance with the question below. Thank you so much.
Setup
Fix $k$ a field of characteristic zero. Let $Sm_{k}$ denote the ...
- 317
2
votes
0
answers
91
views
A sectionwise fiber sequence is homotopy fiber sequence?
Let $\mathscr{C}$ be a site and $\mathsf{sPre}(\mathscr{C})$ the category of simplicial presheaves on $\mathscr{C}$ equipped with Jardine's local model structure. Let $E\to B$ is a sectionwise Kan ...
- 1,836
4
votes
1
answer
187
views
Are simplicial abelian sheaves fibrant?
I want to show that simplicial abelian sheaves are fibrant. For this, I wonder whether a morphism between simplicial sheaves is a fibration iff it has RLP w.r.t. all morphisms like
$$\Lambda^n_k\times ...
- 814
3
votes
0
answers
82
views
Could we have the simplicial definition of equivariant derived category of sheaves with arrow direction inversed?
Let $X$ be a topological space and $G$ be a topological group acting on $X$ from the left. We consider the simplicial space $[G\backslash X]_{\cdot}$ where
$$
[G\backslash X]_n=\underbrace{G\times \...
- 8,627
4
votes
0
answers
235
views
Homotopy colimit description of stacks
Let $F$ be an Artin stack. If $p: X \to F$ is an atlas for $F$, can we express $F$, in the $\infty$-category ${\rm Shv}^{\acute{et}}(k)$ of higher stacks, as a homotopy colimit over the simplicial ...
- 111
2
votes
0
answers
150
views
Is there a definition of an unpointed schematic homotopy type?
In the paper Champs Affines (http://www.math.univ-toulouse.fr/~btoen/chaff.pdf) Toen introduces pointed schematic homotopy types (SHTs) to solve Grothendieck's schematization problem (described in ...
- 1,615
2
votes
0
answers
132
views
Defineing a Sheaf of rings over a topological space
Let $X$ be a topological space and let $R$ be a commutative ring with $1$ such that for each $x\in X$ there exists a multiplicatively closed subset $S_x$ of $R$ such that for each $a\in R$ if $\frac{a}...
- 31
2
votes
0
answers
68
views
Relation of nerve of groupoid and 1st Postnikov object
Let $B$ be a fibrant simplicial set and let $B^{(1)}$ be its 1st Postnikov object. Let $\mathscr{G}$ denote a groupoid such that $\mathrm{Obj}(\mathscr{G})=B_{0}=B^{(1)}_{0}$ and $\mathrm{Aut}_{\...
- 141
7
votes
0
answers
205
views
Pushout of Nisnevich sheaves
Let us consider the projective line $\mathbb{P}^1$ over a field $k$ and take the following open embeddings
$$j_{\epsilon}\colon \mathbb{P}^1\setminus\{0,\infty\} \to \mathbb{P}^1\setminus\{\epsilon\}$$...
- 303
4
votes
0
answers
82
views
$\mathrm{\Gamma}$ functor of Barratt-Eccles in simplicial context
In the article A free group functor for stable homotopy theory, Barratt and Eccles define for each $X\in\mathsf{sSet}_{\ast}$, the free simplicial monoid $\Gamma^{+}X$.
Proposition 6.2 states
if ...
- 795
8
votes
0
answers
185
views
Unaugmentable cosimplicial simplicial sheaves and realization functor
I'm studying the construction of the $\mathrm{Sing}$ functor in Morel-Voevodsky ``$\mathbb{A}^1$-homotopy theory of schemes'' and I was trying to understand the properties of its left adjoint, the ...
- 303
8
votes
1
answer
330
views
An explicit isomorphism between the 1st Cech cohomology and the 1st hypercohomology
Let $\mathbf{X}$ be a Grothendieck topos and let $A$ be an abelian group in $\mathbf{X}$.
Verdier's Theorem allows one to describe $\mathrm{H}^n(\mathbf{X},A)$ in terms of hypercoverings, namely, as ...
- 2,836
4
votes
1
answer
221
views
Matching objects and hypercovers in topology
Question:
Let $X$ be a topological space, $U_*\rightarrow X$ be an augmented simplicial space, and let $M_n(U_*)$ be the n-th matching object computed in $sTop$ while $M_n^X(U_*)$ denotes the $n$-th ...
- 93
3
votes
0
answers
154
views
groupoids representing mapping stacks
1)Let $X$ be a differentiable stack ((2,1) sheaf over the category of smooth manifolds $Man$) and that is geometric. Let $N\in Man$, then
$$
Map(y(N),X)
$$
is again a differentiable stack ($y$ is the ...
- 1,424
3
votes
1
answer
277
views
Basic technical things about simplicial sets to have a good understanding of quasicategories
May someone provide me the list of basic techniques about simplicial sets, in order to have a good understanding of the definition of a quasicategories, $\infty$-topos, $\infty$-stacks, $\infty$-...
- 153
1
vote
0
answers
79
views
Cofibrancy of simplicial objects [duplicate]
Let $\mathcal{C}$ be a site. Consider $sPsh(\mathcal{C})$ be the equipped with the local projective model structure. Let $C_{\bullet}$ be a cofibrant object in $\mathcal{C}$ and let $y(C_\bullet)$ be ...
- 1,424
2
votes
0
answers
202
views
stackifickation of BG
Let $Man$ be the category of smooth manifold. Fix a $M\in Man$ and let $G$ be a group acting smoothly on $M$. The nerve of the group action on $M$ by $G$ defines a simplicial presheaf $X\: : \:Man\to ...
- 1,424
6
votes
2
answers
917
views
Smash product of spheres in $\mathbf{SH}$ and product in cohomology
I have two very concrete and simple question. Just in case I write downwards what led me into this.
My questions: Let $\mathbf{SH}(X)$ be the stable homotopy category of Voevodsky. Denote $S^n$ the ...
- 2,711
1
vote
1
answer
253
views
How to define the internal hom between presheaves valued in cotensored categories?
First let $\mathcal{V}$ be a closed symmetric monoidal category
and $\mathcal{M}$ be a category enriched over $\mathcal{V}$. Moreover we assume $\mathcal{M}$ is cotensored, or powered over $\mathcal{...
- 8,627
1
vote
0
answers
218
views
Internal Hom on simplicial presheaves and the preservation of cofibrant objects
1)Let $\mathcal{C}$ be a cartesian closed small category. Let $\operatorname{Map}\: : \: sPsh(\mathcal{C})\times sPsh(\mathcal{C})\to sPsh(\mathcal{C})$ be the internal Hom of simplicial presheaves, i....
- 1,424
4
votes
1
answer
276
views
Glueing a property via homotopy colimits
I have a problem concerning a fact which is stated without proof in this Rezk's draft: http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf .
In the proof of Lemma 2.11, we are given a ...
- 1,403
1
vote
1
answer
130
views
Does the kan extension preserves contractible presheaves?
Let $\mathcal{C}$, $\mathcal{D}$ be two small categories. Let $f\: : \: \mathcal{C}\to \mathcal{D}$ be a functor. Then it induces a functor
$$
f^{*}\: : \: sPsh(\mathcal{D})\to sPsh(\mathcal{C})
$$
...
- 1,424
6
votes
1
answer
563
views
Descent properties of spaces
I am trying to make sense of what is written in Rezk's draft http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf
In particular, I am referring to Proposition 2.3, which is there stated ...
- 1,403
0
votes
0
answers
210
views
Quillen adjunction betwen simplicial presheaves and cochain complexes
Let $sPsh(\mathcal{C})$ the category of simplicial presheaves over a small category $\mathcal{C}$. Let $Ch^{*}_{\geq 0}$ be the category of positively graded cochain complexes of modules over a field ...
- 1,424
11
votes
1
answer
477
views
Is there a practical criterion to determine whether the limit of a diagram of real chain complexes is also a homotopy limit?
Consider a diagram D: I→ChR of real connective chain complexes.
In the example I have in mind all chain complexes are concentrated in some fixed degree n.
There is a canonical map lim D → holim D ...
- 35.5k
4
votes
0
answers
300
views
Jardine model structure as left Bousfield localization
This should be a really basic question, but I'm stuck on it.
The question. I see written everywhere (for example here, or in the article [DHI] Hypercovers and simplicial presheaves of Dugger, ...
- 503
9
votes
1
answer
540
views
Homotopy left-exactness of a left derived functor
Let
$$
F: \mathcal{C} \leftrightarrows \mathcal{D} :G
$$
be a Quillen adjunction between model categories. Consider the corresponding adjunction of total derived functors
$$
\mathbb{L}F: \mathrm{Ho}(\...
- 3,376
6
votes
0
answers
479
views
Is there a local projective model structure on simplicial sheaves? What are its fibrant objects?
Consider a site S (I am mostly interested in hypercomplete sites, e.g., the site of smooth manifolds).
The category of simplicial presheaves SPSh(S) on S can be equipped with the local projective ...
- 35.5k
11
votes
1
answer
1k
views
Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves
Consider the global projective model category
of simplicial presheaves on some category
(the category of smooth manifolds is particularly interesting to me).
In Section 9.1 of Dugger's paper “...
- 35.5k