Triangular Objects and Systematic K-Theory

We investigate modules over “systematic” rings. Such rings are “almost graded” and have appeared under various names in the literature; they are special cases of the G-systems of Grzeszczuk. We analyse their K-theory in the presence of conditions on the support, and explain how this generalises and unifies calculations of graded and filtered K-theory scattered in the literature. Our treatment makes systematic use of the formalism of idempotent completion and a theory of triangular objects in additive categories, leading to elementary and transparent proofs throughout.

The Evolution of Extreme Cooperation via Shared Dysphoric Experiences

Willingness to lay down one’s life for a group of non-kin, well documented in the
historical and ethnographic records, represents an evolutionary puzzle. Here we
present a novel explanation for the willingness to fight and die for a group, combining evolutionary theorizing with empirical evidence from real-world human groups. Building on research in social psychology, we develop a mathematical model showing how conditioning cooperation on previous shared experience can allow extreme (i.e., life-threatening) pro-social behavior to evolve. The model generates a series of predictions that we then test empirically in a range of special sample populations (including military veterans, college fraternity/sorority members, football fans, martial arts practitioners, and twins). Our results show that sharing painful experiences produces “identity fusion” – a visceral sense of oneness – more so even than bonds of kinship, in turn motivating extreme pro-group behavior, including willingness to fight and die for the group. These findings have theoretical and practical relevance. Theoretically, our results speak to the origins of human cooperation, as we offer an explanation of extremely costly actions left unexplained by existing models.
Practically, our account of how shared dysphoric experiences produce identity fusion, which produces a willingness to fight and die for a non-kin group, helps us better understand such pressing social issues as suicide terrorism, holy wars, sectarian violence, gang-related violence, and other forms of intergroup conflict.