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created differentiation and chain rule
created differentiation and chain rule
These are subjects I’ve spent days and months meditating about and went to the pages expecting to see something familiar. But even as long as I’ve thought about these guys, you still managed to teach me something new. Thanks!
I still haven’t completely grokked it, but it seems like a very nice way to view things. It’d be fun to write a calculus 101 text from this perspective.
It’d be fun to write a calculus 101 text from this perspective.
I think Anders Kock’s textbooks on snthetic differential geometry are meant to achieve a goal like this. They try to re-teach in an elementary fashion calculus and differential geometry.
But it’s a funny thing: now that I understand it, I can read Kock’s book and feel that it nicely accomplishes this. But when years back I started to read it first, I was first very puzzled. I kept oscillating between thinking that, indeed, this is all very simple and considerably simpler than the tranditional way to describe these things, and thinking that I had no idea what kinds of tricks were actually being played with me behind the scenes.
I know the same happened to other people that I talked to.
I believe what had irritated me was that Kock starts writing his book literally from the perspective of inside the topos. What I would have needed back then when I tried to learn this first would have been a precise statement at the beginning along the lines: we work with a category $E$ that has the following structure and property… There is an object called $R$ and an object called $D$ assumed in this category, with the property that suchandsuch.
I know today that he does effectively say that. But nevertheless back then the book didn’t quite serve the calculus-101 purpose for me. Maybe some day I get a chance to teach this myself, and then I could experiment with some hapless students on how to say it instead.
What I would have needed back then when I tried to learn this first would have been a precise statement at the beginning along the lines
But then it wouldn’t be done in an elementary fashion. Nobody begins a standard textbook saying: we work in a topos with a natural numbers object in which every epimorphism splits. They just start working.
However, that would be a nice thing for an aside or an appendix on foundations.
Urs, when Igor was trying to express some ideas by saying that he works in topos, when discussing with us in 2005, you were complaining as to why do you need a topos and things like that; as you had worked with different devices at the time. So maybe you can understand where the complaints of other user come from.
I agree with you as far as the Kock's book gives nice intro into a clean way of organizing differential calculus and applying it to advanced situations like differential geometry and done in a way which is general and not only about the real analysis, but say situations in other fields. However, to do elementary calculus one should understand also the function theory (not only the geometrical origin of calculus), like which functions are diferentiable and which series are convergent. These things are specific and do not come out in my understanding from synthetic geometry. On the other hand, the nonstandard analysis gives an elementary point of view which does those as well, but the differentials over there are not nilpotent and there are many other differences, including the strength of transfer principle. The latter is not only about analysis, the nonstandard analysis can be applied to algebra, set theory and so on, as the nonstandard elements in say internal sets are not just to denote infinitesimals but lots of other situations. Topos theory can be used to model the latter as well in a way which also includes things like forcing (which can not be modeled to my knowledge by synthetic geometry a la Kock).
I started thinking about how this can be extended to more general situations, e.g. possibly finite categories, and remembered your stuff on tangent categories. I think the definition evolved a bit and the version on the nLab is something I no longer recognize.
The abelianization implies (to me, but I could be wrong) that you’ve restricted attention to continuum categories. The earlier (non-abelian) definition seemed to be more general and easily incorporated finite categories.
Is there a way to define a general differentiation as a functor $d: Cat\to Cat$ that sends a category $X$ to its (non-commutative) tangent category $T X$ and functors $f:X\to Y$ to functors $d f: T X\to T Y$?
AND a light bulb just flashed…
This is obvious to everyone, but the usual “$d$ commutes with smooth maps” is simply a statement about the existence of a natural transformation. heart :)
The abelianization implies (to me, but I could be wrong) that you’ve restricted attention to continuum categories.
You are right: you are wrong. ;-)
Is there a way to define a general differentiation as a functor $d: Cat\to Cat$ that sends a category $X$ to its (non-commutative) tangent category $T X$ and functors $f:X\to Y$ to functors $d f: T X\to T Y$?
Yes. That’s called Goodwillie calculus.
The abelianization implies (to me, but I could be wrong) that you’ve restricted attention to continuum categories.
You are right: you are wrong. ;-)
Not surprised :)
But would it be possible to provide a simple example of a nontrivial differentiation + chain rule on a finite category? This is one thing that is kind of ingrained in me, so seeing a counterexample would be very helpful so that I can stop repeating this mistake.
Yes. That’s called Goodwillie calculus.
Neat! Ok. I’ll have a look. Thanks :)
Ok. On Goodwillie calculus, I see the statement:
So now why should spectra/cohomology theories be thought of as linear functors? Well if you think of spectra as analogous to abelian groups, then applying a spectrum to a space (i.e. smashing with it) is a linearization of that space.
At the risk of “strike two”…
I could be wrong, but…
… in order to have a hope of linearizing a space, I’d think that space needs to be a continuum.
By the way, I vaguely see your point about Goodwillie calculus if I squint and accept that $Stab(C)$ is some kind of tangent bundle. Is it?
And for “strike three”…
I could be wrong, but…
The version of tangent category that appears at tangent category involves “abelian group objects” and I suspect the reason for this has something to do with spectra, which then (via the quote above) relates to linearizations, which (via strike two above) relates to a continuum.
So I know I’m on shaky ground (obviously), but I’m still at a loss as to how the current definition of tangent category can involve non-continuum categories in a nontrivial way.
By the way, I tried to go back through the super cool discussion you guys had on tangent categories. At the time, I remember trying to follow, but I had just switched jobs and the financial avalanche had already begun, so I was distracted.
At one point, Urs said:
Let $C$ be any category. Its tangent space at any object $x\in Obj(C)$, which I’ll write
$T_x(C),$ought to be the category whose objects are morphisms in $C$ starting at $x$ and whose morphisms are commuting triangles.
The entire “tangent bundle” of $C$ is then the disjoint union of all these categories
$T C\coloneqq \bigoplus_{x\in Obj(C)} T_x C.$
I’m sure the full story is there, but I think it would be nice to see on the nLab how the notion evolved from this nice and simple (and beautiful if I might add) concept to the one currently residing at tangent category, which I find hard to motivate.
for emphasis I added to chain rule also a remark on how it states the associativity of the composite
$[-1,1] \stackrel{\gamma}{\to} X \stackrel{f}{\to} Y \stackrel{g}{\to} Z$I have started typing at differentiation a section optimistically titled
(this is about the text at differentiation – Exposition of differentiation via infinitesimals)
The spaces $(X^{\Delta^1})_{inf} = X \times D$ and $T X = X^D$ are different. For a more comprehensive discussion of how functions on spaces of infinitesimal simplices are differential forms, see also
(I have added a pointer to that also to the entry).
I renamed one “differential” into “derivative” in the entry. Feel invited to adapt terminology further, where you see the need. I am not dogmatic about this.
It is true that the exposition section on differentiation via infinitesimals does not explicitly connect to the maps between tangent bundles that the Idea-section mentions. If you know how both are related, feel invited to add further discussion to the entry. If you don’t know how both are related, let me know and I can try to find time to add more discussion. (Or maybe somebody else here feels inspired to do so.)
Concerning that relation: by the Kock-Lawvere axiom of SDG we have the standard fact that $T \mathbb{R} := \mathbb{R}^D \simeq \mathbb{R} \times \mathbb{R}$. Now given a function
$\mathbb{R} \times D \longrightarrow \mathbb{R}$this is by the hom-adjunction equivalently a function
$\mathbb{R} \longrightarrow \mathbb{R}^D = T \mathbb{R} \,.$The condition on the original function makes this adjunct be a section of the tangent bundle of $\mathbb{R}$. This section is $x \mapsto f'(x)$, hence is the derivative of $f$ regarded as a tangent vector on $\mathbb{R}$. Now since $\mathbb{R}$ is a microlinear space, this induces by rescaling a function
$T \mathbb{R} = \mathbb{R} \times \mathbb{R} \longrightarrow T \mathbb{R} \,.$And that’s the differential of $f$ regarded as a map of tangent bundles.
Concerning your question on the role of analysis:
It is true that the definition of smooth algebras ($C^\infty$-rings) involves analysis, it enters in the definition of smoothness. But the point of “synthetic” reasoning is that one lists abstract characteristic properties axiomatically as needed for some purpose without picking a particular model. Smooth algebras happen to be one model for the “synthetic” axioms of differential geometry. This model involves analysis, but the abstract framework does not. The wealth of theorems provable from just the “synthetic” axioms serves to show how little they depend on the concrete details of the model. And in fact there are other models of SDG not involving anaylsis. First and formost suitable flavors of algebraic geometry provide models of SDG. (That was Lawvere’s original starting point, he wanted to abstract away all the nice constructions of Grothendieck in algebraic geometry in order to make them more widely available.)
Okay, thanks. It’s good that you spot and point out gaps of exposition in $n$Lab entries. I cannot promise that I’ll have time to react to all such, but I’ll try. And maybe somebody else will chime in, too.
The paragraph from my previous message here I have now pasted into the entry at differentiation – As infinitesimal differences – As a map of tangent bundles.
This would deserve to be further expanded a little. But myself I won’t spend time on this right now.
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