How to define Kahler differential in an abelian category or more general category? Say exact category?
Is there any interesting example?
How to define Kahler differential in an abelian category or more general category? Say exact category?
Is there any interesting example?
It turns out that the whole complex of concepts
and all other aspects of 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
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
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.
There is a paper by Rick Blute, Robert Seely and Robin Cockett on differential categories that partially answers a related if more general question. (It is in Mathematical Structures in Computer Science (2006), 16:6:1049-1083.) They ask for an additive symmetric monoidal category with a comonad and derive a form of differential calculus. (This is linked to Linear Logic so initially will look strange no doubt, but look at their examples.) It is really the dual that is useful in maths, the form as given there is the one more relevant to logic and computer science. Here is a bit of the abstract of that paper:
we introduce the notion of a differential category: an additive symmetric monoidal category with a comonad (a ‘coalgebra modality’) and a differential combinator satisfying a number of coherence conditions. In such a category one should imagine the morphisms in the base category as being linear maps and the morphisms in the coKleisli category as being smooth (infinitely differentiable). Although such categories do not necessarily arise from models of linear logic, one should think of this as replacing the usual dichotomy of linear vs. stable maps established for coherence spaces.
Note Their notion of additive is weaker than the one often used. Their categories are enriched over additive monoids.
In more recent work (in final stages of writing) they (plus myself) show how this relates to the existence of Kahler differentials in quite a general class of cases of the above. (What is still unknown is how to fit Fox derivatives and similar things into the same sort of system.)
I hope this helps.