A Functorial Approach to the Infinitesimal Theory of Groupoids

TL;DR

This paper introduces Nishimura algebroids, a functorial framework for the infinitesimal theory of groupoids, providing a more natural approach than traditional Lie algebroids and connecting to Lie algebroids.

Contribution

It proposes Nishimura algebroids as a new functorial construction for infinitesimal groupoid theory, enhancing the conceptual understanding and linking to Lie algebroids.

Findings

Nishimura algebroids are introduced as a functorial approach.

Totally intransitive Nishimura algebroids are analyzed in detail.

Nishimura algebroids naturally induce Lie algebroids.

Abstract

Lie algebroids are by no means natural as an infinitesimal counterpart of groupoids. In this paper we propose a functorial construction called Nishimura algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids, intended for differential geometry, are of the same vein as Lawvere's functorial notion of algebraic theory and Ehresmann's functorial notion of theory called sketches. We study totally intransitive Nishimura algebroids in detail. Finally we show that Nishimura algebroids naturally give rise to Lie algebroids.