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

Is there any interesting example?

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    $\begingroup$ As posed, the question is so vague (Kähler differentials in what sense? of what object?) as to demand an order of magnitude more work from respondents than from the original question. -1 for being stone soup (which I will revert if enough details are added to give a reader a fighting chance). $\endgroup$ – Yemon Choi Jan 25 '10 at 10:18
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    $\begingroup$ ä - $\endgroup$ – Jonas Meyer Jan 25 '10 at 10:58
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    $\begingroup$ -1. Please read the "how to ask" page. $\endgroup$ – S. Carnahan Jan 25 '10 at 11:20
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    $\begingroup$ Closed, please see the "how to ask" page. Questions need to admit answers that can be reasonably recognised as sufficient. This question is too broad, and asks far more of the respondent than you have contributed yourself. The subject matter may well admit an excellent question, but this isn't the right way to ask it. Feel free to follow up on meta if you disagree with, or don't understand, the reaction you're seeing here. $\endgroup$ – Scott Morrison Jan 26 '10 at 0:49
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    $\begingroup$ I am a bit surprised by the negative reactions here. The question seems to be perfectly reasonable and interesting to me: how do we define Kähler differentials over something other than ordinary rings? As it goes, it happens to be a simple question with a deep answer. I don't think it is wise to discourage simple-sounding questions. Often these are those that get to the heart of the matter. I had that simple question myself a while ago, and was surprised to see the full answer develop eventually. On the nCafe there was recently a long, long disussion about just a tiny aspect of this question. $\endgroup$ – Urs Schreiber Jan 26 '10 at 12:21

It turns out that the whole complex of concepts

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.

  • $\begingroup$ "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." I thought that's in Jon Beck's thesis? $\endgroup$ – Yemon Choi Jan 25 '10 at 22:15
  • $\begingroup$ Ah, I didn't know. That's the first reference? Thanks. I saw it in Quillens "Homotopical algebra". $\endgroup$ – Urs Schreiber Jan 26 '10 at 1:23
  • $\begingroup$ Peter Lee, I just spent a bit of time brushing up the nLab entry on Kähler differentials. Please have a look HERE: ncatlab.org/nlab/show/K%C3%A4hler+differential . This is meant to now pedagogically discuss the pedestrian picture and the grand picture, the concrete examples as well as the sophisticated examples. $\endgroup$ – Urs Schreiber Jan 26 '10 at 1:27
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    $\begingroup$ Urs. Actually, there is a general definition of Kahler differentials in grothendieck sites given by A.Rosenberg in his preprint in Max-Plank. He used stable category to produce deRham complexes even in non additive category $\endgroup$ – Shizhuo Zhang Jan 26 '10 at 2:44
  • $\begingroup$ Thanks for the remark, probably Zoran Skoda told me about this work by Rosenberg before, and I forgot. I'll have another look. But notice that the construction I mentioned also works in non-additive categories. It works, strikingly, for any category whatsoever! Somebody should check how it relates to Rosenberg's construction in cases where that applies. $\endgroup$ – Urs Schreiber Jan 26 '10 at 12:15

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.


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