nLab Beck derivation

Idea

The abstract notion of a derivation corresponding to that of a Beck module.

Definition

Given a category $C$ with finite limits, a Beck module in $C$ over an object $A\in C$ is an abelian group object in the slice category $C/A$.

The forgetful functor from modules to rings is modeled by the forgetful functor

$$U_A \colon Ab(C/A) \longrightarrow C/A \,.$$

Given $M\in Ab(C/A)$, a Beck derivation $A\to M$ is a a morphism $id_A \to U_A(M)$ in $C/A$.

If $U_A$ has a left adjoint $\Omega_A$, then $\Omega_A$ is known as the Beck module of differentials over $A$. Thus, Beck derivations $A\to M$ are in bijection with morphisms of Beck modules

$$\Omega_A\to M,$$

generalizing the universal property of Kähler differentials.

Examples

For ordinary commutative algebras, Beck derivations coincide with ordinary derivations.

For C^∞-rings, Beck derivations coincide with C^∞-derivations.

References

The original definition is due to Jon Beck. An exposition can be found in Section 6.1 of

• Michael Barr, Acyclic models, CRM Monograph Series 17 (2002) [ams:crmm-17, pdf, pdf]