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).
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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 “...
11
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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 ...
9
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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
$$
\mathbb{L}F: \mathrm{Ho}(\...
8
votes
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An explicit isomorphism between the 1st Cech cohomology and the 1st hypercohomology
Let $\mathbf{X}$ be a Grothendieck topos and let $A$ be an abelian group in $\mathbf{X}$.
Verdier's Theorem allows one to describe $\mathrm{H}^n(\mathbf{X},A)$ in terms of hypercoverings, namely, as ...
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Unaugmentable cosimplicial simplicial sheaves and realization functor
I'm studying the construction of the $\mathrm{Sing}$ functor in Morel-Voevodsky ``$\mathbb{A}^1$-homotopy theory of schemes'' and I was trying to understand the properties of its left adjoint, the ...
7
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0
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Pushout of Nisnevich sheaves
Let us consider the projective line $\mathbb{P}^1$ over a field $k$ and take the following open embeddings
$$j_{\epsilon}\colon \mathbb{P}^1\setminus\{0,\infty\} \to \mathbb{P}^1\setminus\{\epsilon\}$$...
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2
answers
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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 ...
6
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1
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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 ...
6
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1
answer
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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 ...
6
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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 ...
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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 ...
5
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0
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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
votes
1
answer
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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 ...
4
votes
1
answer
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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 ...
4
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1
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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 ...
4
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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) ...
4
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0
answers
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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 ...
4
votes
0
answers
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$\mathrm{\Gamma}$ 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 ...
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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, ...
3
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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, $\infty$-...
3
votes
1
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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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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, &...
3
votes
1
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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 ...
3
votes
0
answers
124
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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\...
3
votes
0
answers
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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 ...
3
votes
0
answers
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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
$$
[G\backslash X]_n=\underbrace{G\times \...
3
votes
0
answers
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groupoids representing mapping stacks
1)Let $X$ be a differentiable stack ((2,1) sheaf over the category of smooth manifolds $Man$) and that is geometric. Let $N\in Man$, then
$$
Map(y(N),X)
$$
is again a differentiable stack ($y$ is the ...
2
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1
answer
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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 ...
2
votes
0
answers
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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 ...
2
votes
0
answers
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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 ...
2
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0
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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}...
2
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0
answers
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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}_{\...
2
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0
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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
vote
1
answer
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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
vote
1
answer
260
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{...
1
vote
0
answers
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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 ...
1
vote
0
answers
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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
vote
0
answers
221
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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....
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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 ...
0
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0
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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 ...