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The abstract notion of a derivation corresponding to that of a Beck module.
Given a category C with finite limits, a Beck module in C over an object A∈C is an abelian group object in the slice category C/A.
The forgetful functor from modules to rings is modeled by the forgetful functor
UA:Ab(C/A)→C/A.Given M∈Ab(C/A), a Beck derivation A→M is a a morphism idA→UA(M) in C/A.
If UA has a left adjoint ΩA, then ΩA is known as the Beck module of differentials over A. Thus, Beck derivations A→M are in bijection with morphisms of Beck modules
ΩA→M,generalizing the universal property of Kähler differentials.
For ordinary commutative algebras, Beck derivations coincide with ordinary derivations.
For C^∞-rings, Beck derivations coincide with C^∞-derivations.
The original definition is due to Jon Beck. An exposition can be found in Section 6.1 of
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