We solve a problem proposed by Khovanov by constructing, for any set of primes S, a triangulated category (in fact a stable∞-category) whose Grothendieck group is S−1Z. More generally, for any exact∞-category E, we construct an exact∞-category S−1E of equivariant sheaves on the Cantor space with respect to an action of a dense subgroup of the circle. We show that this∞-category is precisely the...

An important area of business ethics research focuses on how otherwise normal and law-abiding individuals can engage in acts of corruption. Key in this literature is the concept of rationalization. This is where individuals attempt to justify past and future corrupt deeds to themselves and others. In this article, we argue that rationalization often entails a process of overcompensation whereby...

We give an overview of five rationalization theories for spaces (Bousfield-Kan’s ℚ-completion; Sullivan’s rationalization; Bousfield’s homology Casacuberta-Peschke’s Ω-rationalization; Gómez-Tato-Halperin-Tanré’s π1-fiberwise rationalization) that extend the classical simply connected spaces. also corresponding groups (ℚ-completion; Hℚ-localization; Baumslag Malcev completion.

We introduce AI rationalization, an approach for generating explanations of autonomous system behavior as if a human had performed the behavior. We describe a rationalization technique that uses neural machine translation to translate internal state-action representations of an autonomous agent into natural language. We evaluate our technique in the Frogger game environment, training an autonom...

While most Western music today uses the well-established equal-tempered 12 tone scale, there are many reasons why composers and music theorists are still interested in “just” scales in which the scale tones are simple integer ratios. Therefore the rationalization of musical scales is an important problem in music theory. The first general scale rationalization algorithm based on a complete enum...