[Paper Review] A relative notion of algebraic Lie group and applications to $n$-stacks

Carlos Simpson|ArXiv.org|Jul 3, 1996

Homotopy and Cohomology in Algebraic Topology11 references5 citations

TL;DR

This paper introduces 'presentable group sheaves'—a relative generalization of algebraic Lie groups over a base scheme S—extending concepts like connectedness and Lie algebras to relative settings. It uses these to define 'presentable n-stacks,' which are closed under homotopy fiber products and truncation, offering a schematization of homotopy types in characteristic zero, as envisioned by Grothendieck.

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''.

Motivation & Objective

• To generalize the notion of algebraic Lie groups to a relative setting over an arbitrary base scheme S.
• To address A. Grothendieck's quest for schematizing homotopy types in characteristic zero.
• To define a class of n-stacks—'presentable n-stacks'—that are well-behaved under homotopical constructions.
• To extend classical Lie-theoretic notions such as connectedness and Lie algebras to sheaf-theoretic, relative contexts.
• To provide a framework for algebraic geometry to model higher homotopy types via sheaf-theoretic group structures.

Proposed method

• Introduce 'presentable group sheaves' as a full subcategory of sheaves of groups on the big etale site over S, closed under kernels, quotients, and extensions.
• Establish that representable group schemes of finite type over S are presentable, and that pullback and finite direct image preserve presentability.
• Define a relative notion of connectedness for presentable group sheaves, extending the classical notion over Spec(k).
• Construct a Lie algebra object Lie(G) for any presentable group sheaf G, generalizing the classical Lie algebra construction.
• Define 'presentable n-stacks' via conditions on their homotopy sheaves π₀ and πᵢ (for i ≥ 1), requiring π₀ to be representable by a finite-type scheme and πᵢ to be presentable group sheaves.
• Show that the category of presentable n-stacks is closed under homotopy fiber products and truncation, ensuring stability under key homotopical operations.

Experimental results

Research questions

• RQ1 How can the classical notion of an algebraic Lie group over Spec(k) be generalized to a relative setting over an arbitrary base scheme S?
• RQ2 What conditions ensure that a sheaf of groups on the big etale site over S behaves like a Lie group in a relative algebraic geometry context?
• RQ3 Can a notion of schematization of homotopy types be achieved in characteristic zero using sheaf-theoretic group structures?
• RQ4 How do classical Lie-theoretic invariants such as the Lie algebra and adjoint representation generalize in this relative framework?
• RQ5 What structural properties do n-stacks have when their homotopy sheaves are restricted to presentable group sheaves?

Key findings

• Presentable group sheaves form a full subcategory closed under kernels, quotients, and extensions, generalizing finite-type group schemes over S.
• Over S = Spec(k), presentable group sheaves coincide with group schemes of finite type over Spec(k), recovering the classical case.
• A connected presentable group sheaf G satisfies G/Z(G) ≅ G' / Z(G') if Lie(G) ≅ Lie(G'), showing that the adjoint group is determined by its Lie algebra.
• The Lie algebra object Lie(G) exists for any presentable group sheaf G, extending the classical construction.
• Presentable n-stacks are stable under homotopy fiber products and truncation, ensuring good homotopical behavior.
• The framework realizes Grothendieck’s vision of schematizing homotopy types in characteristic zero through sheaf-theoretic generalizations of algebraic Lie groups.