The simplicial-presheaves tag has no usage guidance.

**1**

vote

**1**answer

82 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

**0**answers

69 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

**1**answer

196 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

**1**answer

88 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

**1**answer

289 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

**0**answers

143 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

**1**answer

252 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**

votes

**0**answers

177 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

**1**answer

321 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

**0**answers

204 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

**2**answers

386 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 ...