## Return to Answer

3 added 423 characters in body; added 1 characters in body; edited body; edited body

This is all best explaied by working with groupoids in schemes. By this, I mean, groupoid objects in the category of schemes over $S$. A groupoid object $\mathcal G$ consists of two schemes $\mathcal G_0$ (the objects) and $\mathcal G_1$ (the morphisms) and a whole bunch of maps between them:
• a map $s:\mathcal G_1\to \mathcal G_0$ that sends an arrow to its source.
• a map $t:\mathcal G_1\to \mathcal G_0$ that sends an arrow to its target.
• a map $e:\mathcal G_0\to \mathcal G_1$ that sends an object to its identity arrow.
• a map $i:\mathcal G_1\to \mathcal G_1$ that sends an arrow to its inverse.
• a map $m:\mathcal G_1\times_{\mathcal G_0}\mathcal G_1\to \mathcal G_1$ that composes arrows.
subject to even more axioms.
From now on, I will simply say "groupoid" instead of groupoid abject object in schemes.

Let $G$ be a group over $S$.
Then, there is a groupoid called $EG$, whose objects are $G$, and whose arrows are $G\times_S G$. The groupoid $EG$ is equivelent to $S$, viewed as a groupoid with only identity morphisms. The group $G$ acts freely on $EG$ (this is all happening in the category of schemes over $S$), and the quotient $EG/G$ is $BG$. Here, $BG$ is the groupoid with objects $S$ and morphisms $G$.

If $X$ is a scheme over $S$, then $[X/G]$ is the groupoid whose objects are $X$, and whose morphisms are $X\times _S G$. This groupoid can also be described as the quotient of the (free) diagonal action of $G$ on the groupoid $X\times EG$.

You're asking: "why is the construction of the classifying stack is not related to the notion of the homotopy type, whereas the construction of the classifying space relies on the fact that $EG$ has the homotopy type of a point?".

There are two ways of answering your question:
- One is to make the algebraic-geometric story look a little bit more like what people do in topology. That's what I did above. In particular, the fact that the groupoid $EG$ is equivalent to $S$ is the analog of the fact that $EG$ is contractible in topology.
- The other is to make the topological story a little bit like what people do in algebraic geometry. Namely, instead of defining $BG$ as $EG/G$, define it as the space that represents the functor $X\mapsto$ {iso-classes of $G$-bundles over $X$} where I'm now working in the category of topological spaces and homotopy classes of maps.

I should maybe add:
There is a stack (over the category of $S$-schemes) associated to any groupoid object in $S$-schemes. The converse is not true for general stacks. But it is true, essentially by definition, for Artin stacks. If you restrict yourself to Deligne-Mumford stacks, then you can also assume that the groupoid is pareticularly nice, namely, that the maps $s$, $t$, $e$, $i$ and $m$ are étale.

2 added 796 characters in body; added 2 characters in body

This is all best explaied by working with groupoids in schemes. By this, I mean, groupoid objects in the category of schemes over $S$. A groupoid object $\mathcal G$ consists of two schemes $\mathcal G_0$ (the objects) and $\mathcal G_1$ (the morphisms) and a whole bunch of maps between them:
• a map $s:\mathcal G_1\to \mathcal G_0$ that sends an arrow to its source.
• a map $t:\mathcal G_1\to \mathcal G_0$ that sends an arrow to its target.
• a map $e:\mathcal G_0\to \mathcal G_1$ that sends an object to its identity arrow.
• a map $i:\mathcal G_1\to \mathcal G_1$ that sends an arrow to its inverse.
• a map $m:\mathcal G_1\times_{\mathcal G_0}\mathcal G_1\to \mathcal G_1$ that composes arrows.
subject to even more axioms.
From now on, I will simply say "groupoid" instead of groupoid abject in schemes.

Let $G$ be a group over $S$.
Then, there is a groupoid called $EG$, whose objects are $G$, and whose arrows are $G\times_S G$. The groupoid $EG$ is equivelent to $S$, viewed as a groupoid with only identity morphisms. The group $G$ acts freely on $EG$ (this is all happening in the category of schemes over $S$), and the quotient $EG/G$ is $BG$. Here, $BG$ is the groupoid with objects $S$ and morphisms $G$.

If $X$ is a scheme over $S$, then $[X/G]$ is the groupoid whose objects are $X$, and whose morphisms are $X\times _S G$. This groupoid can also be described as the quotient of the (free) diagonal action of $G$ on the groupoid $X\times EG$.

[Sorry, once again

You're asking: "why is the construction of the classifying stack is not related to the notion of the homotopy type, I pressed whereas the construction of the classifying space relies on the "POST" button too early..fact that $EG$ has the homotopy type of a point?".

There are two ways of answering your question:
- One is to make the algebraic-geometric story look a little bit more like what people do in topology. That's what I did above. In particular, the fact that the groupoid $EG$ is equivalent to $S$ is the analog of the fact that $EG$ is contractible in topology.
- The other is to make the topological story a little bit like what people do in algebraic geometry. Namely, instead of defining $BG$ as $EG/G$, define it as the space that represents the functor $X\mapsto$ {iso-classes of $G$-bundles over $X$} where I'm still writing my answer]now working in the category of topological spaces and homotopy classes of maps.

1

This is all best explaied by working with groupoids in schemes. By this, I mean, groupoid objects in the category of schemes over $S$. A groupoid object $\mathcal G$ consists of two schemes $\mathcal G_0$ (the objects) and $\mathcal G_1$ (the morphisms) and a whole bunch of maps between them:
• a map $s:\mathcal G_1\to \mathcal G_0$ that sends an arrow to its source.
• a map $t:\mathcal G_1\to \mathcal G_0$ that sends an arrow to its target.
• a map $e:\mathcal G_0\to \mathcal G_1$ that sends an object to its identity arrow.
• a map $i:\mathcal G_1\to \mathcal G_1$ that sends an arrow to its inverse.
• a map $m:\mathcal G_1\times_{\mathcal G_0}\mathcal G_1\to \mathcal G_1$ that composes arrows.
subject to even more axioms.
From now on, I will simply say "groupoid" instead of groupoid abject in schemes.

Let $G$ be a group over $S$.
Then, there is a groupoid called $EG$, whose objects are $G$, and whose arrows are $G\times_S G$. The groupoid $EG$ is equivelent to $S$, viewed as a groupoid with only identity morphisms. The group $G$ acts freely on $EG$ (this is all happening in the category of schemes over $S$), and the quotient $EG/G$ is $BG$. Here, $BG$ is the groupoid with objects $S$ and morphisms $G$.

If $X$ is a scheme over $S$, then $[X/G]$ is the groupoid whose objects are $X$, and whose morphisms are $X\times _S G$. This groupoid can also be described as the quotient of the (free) diagonal action of $G$ on the groupoid $X\times EG$.

[Sorry, once again, I pressed on the "POST" button too early...
I'm still writing my answer]
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