Here's a bit of the historical reason why model categories came up. If you have a functor on an abelian category which doesn't quite behave as you would like it to, e.g. is not left exact, there is a ...

One thing the cotangent complex measures is what kind of deformations a scheme has. The precise statements are in Remark 5.30 and Theorem 5.31 in Illusie's article in "FGA explained". Here's the short ...

Here's a simple geometric example for a homotopy pushout. This is stolen from the Dwyer-Spalinski paper on model categories. We first look at the following diagram: pt <-- S^1 --> pt. The pushout ...

Another good place to look are the notes of Master's course on stacks by Betrand Toen. I think they pretty much do exactly what you are looking for. Here's the quick summary: You will want to read ...

I can't give you a complete answer apart from saying that this is definitely a hard problem! If you have a derived scheme $X$, you can always truncate to get an every-day scheme $t_0 (X)$. On the ...

I can't tell you much about the relation to Voevodsky's work, but I can give you a quick summary of Derived Algebraic Geometry. In DAG you enlarge the category of geometric objects that you can ...

I ran across Picard categories in a totally different area of mathematics, but maybe it helps. In short, a Picard category is a group object in the category of groupoids. Picard categories come up ...

In case you are an algebraic geometer and are you used to thinking about a commutative ring in terms of its spectrum it might be helpful to imagine the spectrum of a simplicial commutative ring A as ...

If you work with the naive Chow-groups and allow non-representable morphisms the GRR-Theorem does not hold! In the paper by Toen quoted above and in some of the papers by Joshua there are explicit ...

One of the consequences of flatness of morphisms between projective schemes is that the dimension of the fibers stays constant. Maybe this is the reason for the term. I'm not sure whether this makes ...

Edit: Here is a possible characterization. As mentioned in the comments above, the vanishing of the 1-truncated cotangent complex $\tau_{\leq 1}L_{B/A}$ of a map of rings $f \colon A \to B$ is ...

You can try having a look at this paper: http://arxiv.org/abs/math.AT/0011121 It's the most functorial-minded paper on algebraic geometry I'ver ever seen. It's written by an algebraic topologist. He ...

Zotero is a nice plug-in for firefox that produces a database of your favorite publications. Every time you are on a website like arxiv, math-sci net or the homepage of some journal, it offers you to ...

Another thing you can do is look at the frontispiece of Miles Reid's commutative algebra book (click the frontispiece link in the contents; Google books won't link directly to the right page). It's ...

Here are the two motivations I know of: Number 1 comes from algebraic topology. The definitive reference is http://www.math.harvard.edu/~lurie/papers/survey.pdf , which explains it much better than I ...

In one special case of the Proj construction the functor is easy to write down, namely in the case of $\mathbb{P}^n$. The functor you get in $Sh(Cring^{op})$ is: $R$ gets mapped to split inclusions ...

You might also enjoy this video of a talk by Katrin Wehrheim which compares the approaches a bit. It's actually also quite entertaining. http://www.msri.org/communications/vmath/VMathVideos/VideoInfo/...

Its only a reference, and I don't feel competent to give a summary. I think the answer to question 1 can be found in http://www.math.jussieu.fr/~keller/publ/emalca.pdf . At least it mentions the ...

This is just a little side remark that in general it does make a difference whether you take quotients as representable functors or as schemes. There is a counterexample in section 4 of http://www....