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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeAug 7th 2016
   • (edited Aug 7th 2016)
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   Author: zskoda
   Format: MarkdownItexNew stub [[tangent map]]. It uses the link [[differential of a map]] which does not direct to anything at the moment as it is hard to decide. The entry [[differential]] is dedicated to differential of a chain complex, hence neglecting the term usage for the differential of a map of Banach spaces or the differential of a map of differentiable manifolds. Now the nLab mostly uses [[derivative]] for a differential and at th moment [[derivative]] points to [[differentiable map]]. Now there is an entry [[differentiation]] which is covering mostly the same as [[differentiable map]] but in the way of Lawvere-Kock [[synthetic differential geometry]], Inside the entry [[differentiation]] there is a place whete derivative and differential are contrasted in a way which is exactly opposite to the traditional analysis: the entry calls derivative an infinitesimal difference and differential the ratio, while all classical textbooks do it the opposite to that. Moreover, in that entry, the link [[differential]] is used which points to chain complexes, hence nothing to do (Urs was always complaining that the expression derived functor is not motivated although differentials in chain complexes are used to do it), hence we should not mix differentials in homological algebra and [[differential of a map]], which should become a good redirect, once we agree upon conventions or possible mergers of entries. Attention Urs, Todd, Toby, Mike.

   New stub tangent map.

   It uses the link differential of a map which does not direct to anything at the moment as it is hard to decide. The entry differential is dedicated to differential of a chain complex, hence neglecting the term usage for the differential of a map of Banach spaces or the differential of a map of differentiable manifolds. Now the nLab mostly uses derivative for a differential and at th moment derivative points to differentiable map. Now there is an entry differentiation which is covering mostly the same as differentiable map but in the way of Lawvere-Kock synthetic differential geometry, Inside the entry differentiation there is a place whete derivative and differential are contrasted in a way which is exactly opposite to the traditional analysis: the entry calls derivative an infinitesimal difference and differential the ratio, while all classical textbooks do it the opposite to that. Moreover, in that entry, the link differential is used which points to chain complexes, hence nothing to do (Urs was always complaining that the expression derived functor is not motivated although differentials in chain complexes are used to do it), hence we should not mix differentials in homological algebra and differential of a map, which should become a good redirect, once we agree upon conventions or possible mergers of entries. Attention Urs, Todd, Toby, Mike.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 7th 2016
   • (edited Aug 7th 2016)
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   Author: Todd_Trimble
   Format: MarkdownItexThe term "[[differential]]" as in differential of a chain complex is (I imagine) a back-formation which comes from the example of De Rham complex which is tied directly to the other sense of differential you mention, so I think a brief section on etymology might not be out of order there. I see you also inserted a note of disambiguation at the top, which is good. I haven't yet studied [[differentiable map]] carefully to see how much merging should take place with [[differentiation]]. The article [[differentiation]] looks mostly okay to me, although it could be expanded to include the more traditional point of view, or at least more of a bridge to the traditional point of view. I did not see "differential" described as a ratio anywhere, and my reading of what it says about "derivative" is that it is a *function* that takes infinitesimal differences to infinitesimal differences -- which I think is a correct POV and fits harmoniously with the notion of derivative as ratio (but is conceptually superior: students who learn to think of a derivative as a number have a certain amount of unlearning to do when it comes to rethinking it as rather a linear map on tangent spaces, a coordinate-free description). If I were to write more about that bridge, I think I'd mention that tangent vectors are derivations $\xi: \mathcal{O}_p \to \mathbb{R}$ on local algebras (equivalently, linear forms $\xi: \mathfrak{m}/\mathfrak{m}^2 \to \mathbb{R}$) and that the derivative may be defined by $d f_p(\xi)(\phi) = \xi(\phi \circ f)$ where $\phi \in C^\infty(codomain(f))$-- this description makes functoriality of the derivative manifest. To some extent this is already in the article, but it could be brought out more I think.

   The term “differential” as in differential of a chain complex is (I imagine) a back-formation which comes from the example of De Rham complex which is tied directly to the other sense of differential you mention, so I think a brief section on etymology might not be out of order there. I see you also inserted a note of disambiguation at the top, which is good.

   I haven’t yet studied differentiable map carefully to see how much merging should take place with differentiation.

   The article differentiation looks mostly okay to me, although it could be expanded to include the more traditional point of view, or at least more of a bridge to the traditional point of view. I did not see “differential” described as a ratio anywhere, and my reading of what it says about “derivative” is that it is a function that takes infinitesimal differences to infinitesimal differences – which I think is a correct POV and fits harmoniously with the notion of derivative as ratio (but is conceptually superior: students who learn to think of a derivative as a number have a certain amount of unlearning to do when it comes to rethinking it as rather a linear map on tangent spaces, a coordinate-free description).

   If I were to write more about that bridge, I think I’d mention that tangent vectors are derivations $\xi: \mathcal{O}_p \to \mathbb{R}$ on local algebras (equivalently, linear forms $\xi: \mathfrak{m}/\mathfrak{m}^2 \to \mathbb{R}$) and that the derivative may be defined by $d f_p(\xi)(\phi) = \xi(\phi \circ f)$ where $\phi \in C^\infty(codomain(f))$– this description makes functoriality of the derivative manifest. To some extent this is already in the article, but it could be brought out more I think.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeAug 7th 2016
   • (edited Aug 7th 2016)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI agree with all you said, Todd, you put it quite nicely. Finally, what do you think about my immediate question: to which entry the [[differential of a map]] should point -- [[differentiation]], [[differentiable map]], or elsewhere ? P.S. I added new redirect [[differentiability]] at [[differentiable map]].

   I agree with all you said, Todd, you put it quite nicely. Finally, what do you think about my immediate question: to which entry the differential of a map should point – differentiation, differentiable map, or elsewhere ?

   P.S. I added new redirect differentiability at differentiable map.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 8th 2016
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   Author: Todd_Trimble
   Format: MarkdownItexSomewhat along the lines suggested above, I've added some classically oriented material to [[differentiation]]. (I'd prefer that that table at the end be moved up, somewhere before the reference list as it is more usual for an article to end with references.) I think at this point [[differential of a map]] *could* redirect to [[differentiation]], although I note that [[derivative]] redirects to [[differentiable map]]. I still have yet to have a good look at the latter. It seems a little odd having two lengthy and such closely related articles.

   Somewhat along the lines suggested above, I’ve added some classically oriented material to differentiation. (I’d prefer that that table at the end be moved up, somewhere before the reference list as it is more usual for an article to end with references.)

   I think at this point differential of a map could redirect to differentiation, although I note that derivative redirects to differentiable map. I still have yet to have a good look at the latter. It seems a little odd having two lengthy and such closely related articles.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeAug 8th 2016
   • (edited Aug 8th 2016)
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   Author: Mike Shulman
   Format: MarkdownItexPersonally, I've never been very happy that the page named [[differential]] is about differentials in chain complexes, especially because it's very stubby and I have a hard time imagining what might go there that wouldn't fit just as well at [[chain complex]]. Whereas I think there is a lot to say about the [[differential of a map]] that is distinct from a [[derivative]]. So I would be in favor of taking over the page name [[differential]] for "differential of a map" and redirecting existing (correct) links to point to [[chain complex]] instead. I don't have an informed opinion yet about how much merging should happen between [[differentiable map]] and [[differentiation]], but my initial reaction would be "quite a lot".

   Personally, I’ve never been very happy that the page named differential is about differentials in chain complexes, especially because it’s very stubby and I have a hard time imagining what might go there that wouldn’t fit just as well at chain complex. Whereas I think there is a lot to say about the differential of a map that is distinct from a derivative. So I would be in favor of taking over the page name differential for “differential of a map” and redirecting existing (correct) links to point to chain complex instead.

   I don’t have an informed opinion yet about how much merging should happen between differentiable map and differentiation, but my initial reaction would be “quite a lot”.