Belian categories (Q2827345)

In recent years, the \(\mathbb{F}_1\)-geometry has been developing rapidly. The author writes that ``there are many different approaches to \(\mathbb{F}_1\)-geometry, but their common core seems to be the ``non-additive geometry'' which is a version of algebraic geometry not based on rings, but on monoids. The general idea of that theory is to ``forget addition'' and work with multiplication alone as long as feasible''. The author notes that there are many different generalizations of homological algebra, but none of them is suitable for non-additive geometry.NEWLINENEWLINEThe author introduces a new concept, which he calls ``belian category''. The concept of belian category is necessary in order to study the \(\mathbb{F}_1\)-geometry by homological methods. A belian category is a balanced pointed category \(\mathcal{B}\) which contains finite products, kernels, and cokernels, and has the property that every morphism with zero cokernel is an epimorphism. Every belian category is abelian. The first part of the paper under review is devoted the foundation on homological algebra for non-additive categories. The author explores, in particularly, the extensions in belian categories and the derived category of a belian category. He proves also that the cohomology can be computed using flabby resolutions. In the second part, the author verifies the conditions in the context of sheaves over monoid schemes.NEWLINENEWLINEFor the entire collection see [Zbl 1343.14002].