Tangent categories provide the categorical foundations for (synthetic) differential geometry. Briefly, objects in a tangent category can be thought of as abstract smooth manifolds, each of which has an associated abstract tangent bundle. Tangent categories have been successful is formalizing numerous concepts in differential geometry such as vector fields, vector bundles, differential forms, etc. Tangent categories have also found connections to other areas. In particular, recently there has been interest in studying tangent categories for algebra (where the tangent bundle is given by dual numbers) and algebraic geometry (where the tangent bundle is given by Kahler differentials). In this talk, I will give a friendly introduction to tangent categories, and give some recent interesting results in relation to algebra, algebraic geometry, and operads. This talk will be based on recent work:

https://arxiv.org/abs/2303.05434