Let $G$ be an algebraic group, $X \in \underline{Schm}$ be a scheme, we define the classifying space for G-torsors by
$Mor(X,BG)= \{\text{G-torsors on X \}/ \simeq$.
From the viewpoint of algebraic topology you can take:
$[X,BG]/\text{homotopy} = \{\text{G-bundles}\}/\simeq$.
Let $\underline{BG}: \underline{Schm} \rightarrow \text{Groupoid}, \quad X \rightarrow \{\text{G-Torsors},\simeq\}$, by groupoid i will mean a small category where all arrow are Iso.
$\underline{BG}(X)$ is category with obj = G-torsors, mor = Iso.
Def of stack: that is a 2-funtor $\mathcal{M}$ from Schemes to Groupoids (= 2-Cat), s.t. $\mathcal{M}(S)$ is groupoid.
The 2-functor takes Morphism/schm ( S---> T) ----> ($\mathcal{M}(T) \rightarrow \mathcal{M}(S), \quad \mathcal{M}(f) = f^*$ (induced map by BG).
The composition of Morphism/schm (S--f-->T---g-->W) -----> f* g* => (gf)*. That will give us 2-Morphisms between Morphisms.
Obj ---> 1-Morph ----> 2-Morph. Groupoid -----> Functor ----> natural Transformation of Funct.
Examples: Ví dụ điển hình nhất từ topo là take {Top Spaces}, continous maps and Homotopy. (tổng quát lên là \infty-category, which is extrem interesting obj. from higher topois).
some geom: Schemes define Stacks, which means the glueding property of objects.
Take the analogy to Yoneda ---> you will get somehow morphisms between stacks $F: \mathcal{M} \rightarrow \mathcal{N}$. We call F representable iff for all X scheme or algebraic Space (in sense of Artin) you have:
$\mathcal{M} \times_{\mathcal{N}} X \rightarrow X $ is a morph of Schm or alg Sp. (in sense of Artin).
Def: We call M an Artin stack <=> first: Diagonal: M ---> M x M is representable, quasi-compact and separated.
second: tồn tại 1 atlas Y ---> M smooth and surjective.
Edited by Alexi Laiho, 20-11-2007 - 09:07.