A relative notion of algebraic Lie group and applications to $n$-stacks

TL;DR

This paper introduces presentable group sheaves and presentable n-stacks over schemes, generalizing algebraic Lie groups and addressing Grothendieck's schematization of homotopy types in characteristic zero.

Contribution

It defines presentable group sheaves and n-stacks, extending algebraic Lie groups to a relative setting over arbitrary schemes, and connects these concepts to homotopy type schematization.

Findings

Presentable group sheaves include group schemes of finite type over S.

The category of presentable group sheaves is closed under kernels, quotients, and extensions.

Presentable n-stacks are constructed with sheaves of presentable group sheaves as homotopy sheaves.

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''.