2025/04/29 by Lee, Jaehyeok
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.20949
We establish a generalized version of the duality between groups and the categories of their representations on sets. Given an abstract symmetric monoidal category K called Galois prekosmos, we define pre-Galois objects in K and study the categories of their representations internal to K. The motivating example of K is the cartesian monoidal category Set of sets, and pre-Galois objects in Set are groups. We present an axiomatic definition of pre-Galois K-categories, which is a complete abstract characterization of the categories of representations of pre-Galois objects in K. The category of covering spaces over a well-connected topological space is a prototype of a pre-Galois Set-category. We establish a perfect correspondence between pre-Galois objects in K and pre-Galois K-categories pointed with pre-fiber functors. We also establish a generalized version of the duality between flat affine group schemes and the categories of their linear representations. Given an abstract symmetric monoidal category K called Grothendieck prekosmos, we define what are pre-Grothendieck objects in K and study the categories of their representations internal to K. The motivating example of K is the symmetric monoidal category Veck of vector spaces over a field k, and pre-Grothendieck objects in Veck are affine group k-schemes. We present an axiomatic definition of pre-Grothendieck K-categories, which is a complete abstract characterization of the categories of representations of pre-Grothendieck objects in K. The indization of a neutral Tannakian category over a field k is a prototype of a pre-Grothendieck Veck-category. We establish a perfect correspondence between pre-Grothendieck objects in K and pre-Grothendieck K-categories pointed with pre-fiber functors.