I am trying to figure out whether the concepts of connection and flat connection can be defined in a way analogous to how derivations are defined here by Akhil Mathew.
As is discussed at the link, derivations can be understood using free abelian group objects in overcatevgories of algebras. The tangent bundle can also be understood this way, using the case of an over-object whose structure map is the identity.
Meanwhile, connections have a quite similar property to derivations- one resembling an "analogue for actions". My question is, is there a similar adjunction for connections and an analogue of the cotangent complex which produces a module of connections instead of a module of derivations?
The k-linear dual of the left adjoint of this functor should produce the k-module of connections in the case of the real numbers and rings of smooth sections of a smooth manifold.
The right adjoint here could feature a product with a square zero multiplication, in keeping with the analogy between derivations and connections.