Title: Lie integration as a left Kan extension
Abstract: Charles Ehresmann first introduced Lie groupoids as groupoids in the category of smooth manifolds through his initial work in sketch theory. His student Pradines developed Lie algebroids as the “infinitesimal approximation” of a Lie groupoid (extending the Lie group-Lie algebra correspondence). The question of what Lie algebroids could be “integrated” into a source-simply-connected Lie groupoid was one of the significant problems in differential geometry and Lie theory, and was resolved by Crainic and Fernandes in their paper “On the integrability of Lie brackets.”
In this talk, we will use Kelly’s enriched sketches to show that Lie integration is a left Kan extension. The construction follows in three steps: we first show that Lie algebroids are equivalent to involution algebroids, we next show that involution algebroids are sketchable, and finally we show that the theory of involution algebroids has a fully-faithful, left-exact inclusion into the theory of smooth groupoids. A useful consequence of this result is that in presentable tangent categories (such as models of synthetic differential geometry) we can see that involution algebroids are a coreflective subcategory of smooth groupoids.
This talk is based on joint work with Matthew Burke.