Let $R$ be a commutative ring and $\text{Ch}(R)$ the category of chain complexes of $R$-modules. $\text{Ch}(R)$ is first of all an abelian category, but it can also be equipped with the structure of a (symmetric?) closed monoidal category: it has
- a tensor product $A \otimes B$ whose component in degree $n$ is $\bigoplus_{i+j=n} A_i \otimes_R B_j$ and where the differential is defined by $d(a \otimes b) = da \otimes b + (-1)^{|a|} a \otimes db$, and
- an internal hom $\text{hom}(A, B)$ whose component in degree $n$ is $\prod_i \text{Hom}_R(A_i, B_{i+n})$ and where the differential is defined by $(df)(a) = d(fa) - (-1)^{|f|} f(da)$
which are related via the adjunction $\text{Hom}(A \otimes B, C) \cong \text{Hom}(A, \text{hom}(B, C))$.
How can I motivate the sign convention in either of these definitions?
I can sort of answer that question: both sign conventions are just the graded Leibniz rule. As Theo Johnson-Freyd explains on MO, this is because $\text{Ch}(R)$ is precisely the category of modules over a one-dimensional graded-commutative Lie algebra over $R$ concentrated in degree $1$. But this is unsatisfying because I don't know the answer to this question:
What do graded Lie algebras have to do with the topological motivation for studying chain complexes?
By "the topological motivation" I mean the idea, made precise by the Dold-Kan correspondence, that a chain complex is, roughly speaking, a linearization of a combinatorialization of a topological space. I think that the sign convention for the tensor product can be motivated by thinking about simplicial chains, but this doesn't have quite the air of inevitability that I would like out of so fundamental a definition.
I have the feeling that everything goes back to Euler characteristic. A bounded complex can be sent to an alternating sum of elements of $K_0(R)$, and this sum is invariant under chain homotopy and additive in short exact sequences and possibly satisfies a universal property of some kind, and it clearly shows there is a real distinction to be made between the even and odd elements of a chain complex. But I don't know the conceptual route from this idea to the graded Leibniz rule.