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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJan 8th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## 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 $$U_A\colon Ab(C/A)\to C/A.$$ 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), [PDF](https://www.math.mcgill.ca/barr/papers/acmod.pdf). <a href="https://ncatlab.org/nlab/revision/Beck+derivation/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Beck+derivation">current</a>

   Created:

   Idea

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

   Definition

   Given a category with finite limits, a Beck module in over an object is an abelian group object in the slice category .

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

   Given , a Beck derivation is a a morphism in .

   If has a left adjoint , then is known as the Beck module of differentials over . Thus, Beck derivations are in bijection with morphisms of Beck modules

   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), PDF.

   v1, current