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Hi isomorphism,
this is a misunderstanding. Just because both the de Rham complex, the singular simplicial complex and other complexes are (or give rise to) chain complexes, does not mean that “calculus is topology”. Not at all.
The topic behind all this is homological algebra, it has plenty of applications thoughout mathematics. You’ll be surprised how many things in the world are controled by the equation $d^2 =0$. It’s way, way more than just certain complexes that appear in calculus and topology.
Moreover, this equation is itself just a simple special case, namely the abelian case of something even more profound: the simplicial identities, which you can think of as being the “nonabelian generalization” of $d^2 = 0$.
And this is not even the end of the story either, not by far. There are a bunch of pretty general mechanisms at work behind all of mathematics, and you’ll eventually discover more of them.
There is a point made there which is correct. Given a smooth manifold $X$, if you know its smooth $k$-chains (the formal linear combinations of smooth maps from the smooth $k$-simplex, modulo an equivalence relation that can be described combinatorially) and the boundary operators (also combinatorial) between these, then you can define smooth exterior differential $k$-forms and their differentials as the dual complex. (Of course, this begs the question of what are smooth manifolds and smooth maps.) In particular, if you know what a smooth map from the unit interval to the real line is, then you can define the ordinary derivative of such in this way. This gives some insight; it essentially makes the fundamental theorem of calculus into an axiom.
Moreover, this equation is itself just a simple special case, namely the abelian case of something even more profound: the simplicial identities, which you can think of as being the “nonabelian generalization” of $d^2 = 0$.
As someone who thinks $d^2=0$ is one of the most profound things ever, my ears perk up if someone says there is something even more profound. How are the simplicial identities nonabelian generalizations of $d^2=0$ and are these more profound than globular identities or cubical identities? I’m not quite ready to dethrone my beloved $d^2 = 0$ so easily, but am very curious to hear reasons why I should consider it.
@Eric
in the cases where we use $d^2 = 0$, then the (co)chain complex it belongs to underlies, via the Dold-Kan correspondence (actually an equivalence of categories in some higher sense), a (co)simplicial abelian group (or possibly module etc). In the more general case of a (co)simplicial group (not necc. abelian) then this correspondence is not so nice - it can be done and one gets what are called hypercrossed complexes (this was Pilar Carrasco’s PhD thesis). But the simplicial version is easier. Now simplicial groups correspond to pointed connected homotopy types aka $\infty$-groupoids, in a way that can be made very precise by work of Kan (and generalised to other $\infty$-toposes by recent work by Urs and collaborators).
In more prosaic terms, proving $d^2 = 0$ for a chain complex arising from a simplicial abelian group actually depends on the simplicial identities, and uses the geometric structure of simplices in a very nice way. But this really does require the simplicial group to be abelian, so it is the simplicial identities which are more fundamental.
Thanks David. Like I mentioned, I am very interested in this topic and the suggestion $d^2 = 0$ is less fundamental than simplicial identities is something I would find to be very fascinating. I am certainly open to the possibility, but the example provided is not completely convincing because it seems I could turn the logic around.
Starting with $d^2=0$, I think you can deduce the simplicial identities and more. In other words, imposing $d^2=0$ induces relations among elements. When the elements are simplices, the relations turn out to be simplicial identities. When the elements are different, you get different identities, but the fundamental thing driving this is $d^2=0$. This was important in some earlier work I did with Urs and is also important more generally with universal graded differential algebras and their quotients.
I made a note of what has been said in this thread in the section Meaning in higher category theory in chain complex. A propos, I like the section-title ”Meaning in higher category theory” - in fact in many articles this is an implicit subsection of the ”Idea”-section.
@Eric - but when you move out of the abelian setting, you cannot derive [come of the] simplicial identities from d^2 = 0 because you just don’t have the latter! (imagine trying to claim the identity gh = hg is prior the identity gh = [g,h]hg, because in abelian groups they are equivalent) But! the identity ’d^2 = 0’ only makes use of the face maps, not the degeneracy maps. You cannot (or I suspect very strongly you cannot) derive the identities involving the degeneracy maps from ’d^2=0’.
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