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 $\pi_0$ (the actual condition on $\pi_0$ is slightly more subtle), and if its $\pi_i$ (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''.