The 2-group of linear auto-equivalences of an abelian category and its Lie 2-algebra

TL;DR

This paper constructs a stack of 2-groups associated with an abelian category over a scheme and shows its tangent space forms a Lie 2-algebra, linking category theory and higher algebraic structures.

Contribution

It introduces a stack of 2-groups for abelian categories satisfying (AB5) over schemes and describes its tangent Lie 2-algebra structure, providing new insights into higher categorical symmetries.

Findings

Construction of the stack GL() over the flat site of S

Concrete description of GL() for quasi-coherent sheaves on a scheme

Identification of the tangent space as a Lie 2-algebra

Abstract

For any abelian category \calC satsifying (AB5) over a separated, quasi-compact scheme S, we construct a stack of 2-groups \GL(\calC) over the flat site of S. We will give a concrete description of \GL(\calC) when \calC is the category of quasi-coherent sheaves on a separated, quasi-compact scheme X over S. We will show that the tangent space \gl(\calC) of \GL(\calC) at the origin has a structure as a Lie 2-algebra.