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2
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1answer
116 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, ...
1
vote
0answers
63 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 ...
2
votes
0answers
148 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 ...
5
votes
2answers
221 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 ...
1
vote
1answer
95 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 ...
1
vote
0answers
81 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, ...
4
votes
1answer
208 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
vote
1answer
94 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 $$ f^{*}\: : \: sPsh(\mathcal{D})\to sPsh(\mathcal{C}) $$ ...
5
votes
1answer
301 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 ...
0
votes
0answers
154 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 ...
7
votes
1answer
308 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 ...
3
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0answers
187 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, ...
7
votes
1answer
338 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: ...
4
votes
0answers
219 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 ...
6
votes
2answers
417 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 ...