Let $R$ be a commutative ring. For awhile I have been trying to motivate to myself more fully the definition of and various structures on the category $\text{Ch}(R)$ of chain complexes of $R$-modules (and various subcategories thereof). One significant piece of motivation is the Dold-Kan correspondence, at least when $R = \mathbb{Z}$, which tells us that studying connective chain complexes is like studying linearized homotopy theory (or linearized higher category theory). This is a great idea, but I don't have much intuition for what's going on in the proof of Dold-Kan, and I don't see how one could have predicted in advance that something like Dold-Kan might be true just by looking at all the definitions in the right way. I like the idea of linearized homotopy but I don't know what the conceptual path is from linearized homotopy to, for example, the braiding $a \otimes b \mapsto (-1)^{|a| |b|} b \otimes a$.
Consider also the differential. I can think of various ways to motivate $d^2 = 0$, and I don't quite know how they fit together. For example, one can talk about boundaries of manifolds with boundary, the exterior derivative, and Stokes' theorem. If one starts from the simplicial / higher-categorical perspective, the differential encodes something like the generalized source / target of a higher morphism, and somehow the fact that this generalized source / target ought to satisfy a natural "gluing law" (for example if $a \to b$ is a $1$-morphism then $d(a \to b + b \to c)$ ought to equal $d(a \to c)$) is equivalent to it squaring to zero. I can sort of picture how this works in low dimensions but I don't completely grasp what the exact relationship between these two ideas is.
Keeping in mind the symmetric monoidal structure, the differential behaves like an element of a super Lie algebra, concentrated in degree $-1$, acting on a representation (see for example Theo Johnson-Freyd's MO answer here). The action of a super Lie algebra should be related to infinitesimal symmetry coming from a super Lie group, but I don't have a clear idea of what this super Lie group is or what it has to do with homotopy theory. This seems to have something to do with the supergeometric definition of differential forms, but I don't really know anything about this.
Algebraically, the relation $d^2 = 0$ seems to come from at least two different ideas: first-order approximation, and odd things anti-commuting with themselves. Both of these ideas seem relevant to what I'm confused about, but I can't put them together into a cohesive story.
So what is that cohesive story?
Edit: If the bulk of the question seems sort of silly to you, feel free to focus on that last bit about super Lie algebras. I remember hearing that this has something to do with the action of the automorphism group of the odd real line; I would appreciate if someone could clarify that for me.
Best Answer
$ d^2 $ and homology are almost the most basic and natural quests in mathematics. In general, the image of a map can be thought of as "things that can be constructed". In general, the kernel of a map is "things we can test". So $ \ker d = \operatorname{im }d$ is one of the basic quests in mathematics, linearized: "let us find a construction for all the things that satisfy a certain criterion".
Likewise, given a construction for things that satisfy a certain property (name, given $A\stackrel{d_1}{\longrightarrow}B\stackrel{d_2}{\longrightarrow}C$, where $d_1$ is the construction and $d_2=0$ is the property), the homology $\ker d_2/\operatorname{im }d_1$ measures how successful you had been - to what extent you were able to construct all the things you wanted to construct. Again a very basic mathematical quest.
So two-step complexes, $d^2=0$, and homology are as natural as anything. What puzzles me is that way too often two-step complexes have a natural extension to become many-step complexes. I have no good philosophical explanation for why that should be the case.