How to define Kahler differential in an abelian category or more general category? Say exact category?


Is there any interesting example?



How to introduce Kahler differential in category?
It turns out that the whole complex of concepts


* [module](http://ncatlab.org/nlab/show/module)
* [derivation](http://ncatlab.org/nlab/show/derivation)
* [Kähler differentials](http://ncatlab.org/nlab/show/K%C3%A4hler+differential)
* [cotangent complex](http://ncatlab.org/nlab/show/cotangent+complex)
* [quasicoherent sheaf of modules](http://ncatlab.org/nlab/show/quasicoherent+sheaf)
* and all other aspects of [deformation theory](http://ncatlab.org/nlab/show/deformation+theory)


works in remarkable generality on pure category-theoretic grounds with respect to *any* category, and is nothing but different facets of one single general concept: that of the


* [tangent $(\infty,1)$-category](http://ncatlab.org/nlab/show/tangent+(infinity%2C1)-category).


This goes back to the old observation by Quillen, that the category of modules over a ring is equivalent to the category of abelian group objects in the given overcategory of rings. All other concepts follow from this: derivations are sections through the over-objects, and the assignment of Kähler differentials is the left-adjoint to the projection down from the overcategory.


The notion of "tangent $(\infty,1)$-category" takes this idea to its full generality: this is the over-$(\infty,1)$-category, fiberwise stabilized. See the above link for details.


This complete picture, based on Quillen's old idea, is fully developed and exposed in the first part of the very nice article


* Jacob Lurie, [Deformation Theory](http://ncatlab.org/nlab/show/Deformation+Theory).


So the answer to the question is: a notion of Kähler differentials exists with respect to *any* (oo,1)-category $C$! Here for given $C$, the resulting notion models universal modules for objects in $C^{op}$, regarded as function rings over the objects in $C$.


I can't quite tell what the abelian category is supposed to be that appears in the question. But notice that the plain vanilla version of the story is obtained by letting $C$ be the category of (simplicial) monoids in the abelian category $Ab$ of abelian groups. 


So, indeed, for any abelian category whatsoever, it makes very good sense to regard the category of monoids inside it as a replacement for the category of rings, regard the category of abelian group objects in the slice-categories of that as the corresponding bifibration of modules, and take the corresponding Kähler differentials to be the corresponding universal modules with respect to derivations, just following the general nonsense linked to above.


