A relative notion of algebraic Lie group and applications to n -stacks
Abstract
If
S
is a scheme of finite type over $k=\cc$, let $\Xx /S$ denote the big etale site of schemes over
S
. We introduce {\em presentable group sheaves}, a full subcategory of the category of sheaves of groups on $\Xx /S$ which is closed under kernel, quotient, and extension. Group sheaves which are representable by group schemes of finite type over
S
are presentable; pullback and finite direct image preserve the notions of presentable group sheaves; over
S=Spec(k)
then presentable group sheaves are just group schemes of finite type over
Spec(k)
; there is a notion of connectedness extending the usual notion over
Spec(k)
; and a presentable group sheaf
G
has a Lie algebra object
Lie(G
. If
G
is a connected presentable group sheaf then
G/Z(G)
is determined up to isomorphism by the Lie algebra sheaf
Lie(G)
.
We envision the category of presentable group sheaves as a generalisation relative to an arbitrary base scheme
S
, of the category of algebraic Lie groups over
Spec(k)
.
The notion of presentable group sheaf is used in order to define {\em presentable
n
-stacks} over $\Xx$. Roughly, an
n
-stack is presentable if there is a surjection from a scheme of finite type to its
π
0
(the actual condition on
π
0
is slightly more subtle), and if its
π
i
(which are sheaves on various $\Xx /S$) are presentable group sheaves. The notion of presentable
n
-stack is closed under homotopy fiber product and truncation.
We propose the notion of presentable
n
-stack as an answer in characteristic zero for A. Grothendieck's search for what he called ``schematization of homotopy types''.