Homological algebra in characteristic one

Alain Connes, Caterina Consani

Abstract


This article develops several main results for a general theory of homological algebra in categories such as the category of idempotent modules. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for abelian groups. Our selected prototype, the category $\bmod$ of modules over the Boolean semifield $\B:=\{0,1\}$ is the replacement for the category of abelian groups. We show that the semi-additive category $\bmod$ fulfills analogues of the axioms AB1 and AB2 for abelian categories. By introducing a precise comonad on $\bmod$ we obtain the conceptually related Kleisli and Eilenberg-Moore categories. The latter category $\b2$ is simply $\bmod$ in the topos of sets endowed with an involution and as such it shares with $\bmod$ most of its abstract categorical properties. The three main results of the paper are the following. First, when endowed with the natural ideal of null morphisms, the category $\b2$ is a semi-exact, homological category in the sense of M. Grandis. Second, there is a far reaching analogy between $\b2$ and the category of operators in Hilbert space, and in particular results relating null kernel and injectivity for morphisms. The third fundamental result is that, even for finite objects of $\b2$, the resulting homological algebra is non-trivial and gives rise to a computable Ext functor. We determine explicitly this functor in the case provided by the diagonal morphism of the Boolean semiring into its square.


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