Tangentads: a formal approach to tangent categories

Abstract

Tangent category theory is a well-established categorical context for differential geometry. In a previous paper, a formal approach was adopted to provide a genuine Grothendieck construction in the context of tangent categories 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 tangent (split) restriction 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 tangent split restriction categories are $2$-equivalent to tangent $\mathscr{M}$-categories.