The formal theory of tangentads I

Abstract: Tangent category theory is a well-established categorical framework for differential geometry. In a previous paper, a formal approach was adopted to provide a genuine Grothendieck construction in the context of tangent category theory, by introducing tangentads. A tangentad is to a tangent category as a formal monad is to a monad of a category. In this paper, we discuss the formal notion of tangentads, construct a $2$-comonad structure on the $2$-functor of tangentads, and introduce cartesian, adjunctable, and representable tangentads. We also reinterpret the subtangent structure with negatives of a tangent structure as a right Kan extension. Furthermore, we present numerous examples of tangentads, such as (split) restriction tangent categories, tangent fibrations, tangent monads, display tangent categories, and infinitesimal objects. Finally, we employ the formal approach to prove that every tangent monad admits the construction of algebras, provided the underlying monad does, and show that split restriction tangent categories are $2$-equivalent to $\mathscr{M}$-tangent categories.