Reason to apply the Koszul sign rule everywhere in graded contexts I'm copy-pasting this question I asked in MO that received no answer. 
The Koszul sign rule is a sign rule that arises from graded commutative algebras. For instance, let $\bigwedge(x_1,\dots, x_n)$ be the free graded commutative algebra generated by $n$ elements of respective degrees $\lvert x_i\rvert$. Then, the sign $\varepsilon(\sigma)$ of a permutation $\sigma$ on $(x_1,\dotsc, x_n)$ is given by
$$x_1\wedge\dotsb\wedge x_n=\varepsilon(\sigma)x_{\sigma(1)}\wedge\dotsb\wedge x_{\sigma(n)},$$
which comes from the fact that in a graded commutative algebra one has by definition $a\wedge b = (-1)^{\lvert a\rvert\lvert b\rvert}b\wedge a$. 
There is also an antisymmetric Koszul sign rule which arises from graded anticommutative algebras and it's just the previous sign times the sign of the permutation. Both signs are used for instance in Lada and Markl - Symmetric brace algebras.
However, I've been seeing the Koszul sign rule used in any graded context and even for operations that are not products in some algebra. For example, from Roitzheim and Whitehouse - Uniqueness of $A_\infty$-structures and Hochschild cohomology, given graded maps of graded algebras $f,g:A\to B$, if we want to evaluate $f\otimes g$ in an element $x\otimes y$, apparently we need to apply the sign rule to get 
$$(f\otimes g)(x\otimes y)=(-1)^{\lvert x\rvert\lvert g\rvert}f(x)\otimes g(y),$$
but I see no mathematical reason to do that, it just seems to be a convention.
A more complex example of application of the Koszul sign rule is in the definition of brace algebra (also in the Lada and Markl paper).
I could give many more examples. In some of them I can understand the reason. For instance, the differential of a tensor product of complexes $C$ and $D$ cannot simply be $d_C\otimes 1_D+ 1_C\otimes d_D$ (it can be defined this way if we use the sign rule when we apply it to elements, but in any case it needs the sign). But maps in general need not be differentials. In other cases, the signs appear in nature and one use this sign rule to justify them, as in $A_{\infty}$-algebras, but this feels too artificial for me and doesn't really explain why we should use that sign rule.
So, in the end, every time there is a sequence $(x_1,\dotsc, x_n)$ of graded objects of any kind and not necessarily all of them of the same kind (elements, maps, operations, …), and related in any way (they can be multiplied, or applied, or whatever), we use the Koszul sign rule to permute the sequence. 

To me all of this seems more philosophical than mathematical, and as I said it feels to be just a convention. But, is there some general mathematical reason to use the sign rule in such an extensive way? And if it's just a convention, why should we use it? From my experience, it gets very messy when it comes to applying the sign rule to larger formulas, and in the end everything is just a $+$ or $-$ sign, so I see no advantage.

 A: First of all I should say that I am not familiar with those cohomology theories and high-tech gadgets you mentioned in your problem statement. But recently I learn some graded algebra stuff, and I would like to write what I know about this sign rule. Given two graded abelian groups $A=\{A_m\}_{m\in M}, B=\{B_m\}_{m\in M}$ where $(M, +, 0)$ is a monoid, the graded tensor product is $$A\otimes B=\left\{\bigoplus_{i+j=m}A_i\otimes B_j\right\}_{m\in M}.$$ Most widely-used examples are $M=\mathbb{N}, \mathbb{Z}, \mathbb{Z}_2.$ In theses cases, the graded brading is an isomorphism $$A\otimes B\xrightarrow{\cong}B\otimes A$$ given index-wise by $A_i\otimes B_j\xrightarrow{(-1)^{i+j}}B_j\otimes A_i,$ which is exactly the Koszul sign rule. This rule appears in

*

*cup product operation on cocycles :  $\alpha^p\smile\beta^q=(-1)^{pq}(\beta^q\smile\alpha^p)$

*wedge product of differential forms: $\alpha\wedge\beta=(-1)^{\text{deg}(\alpha)\text{deg}(\beta)}\beta\wedge\alpha$

*super geometry (including Grassmann algebra), etc.

As far as I understand, and as the second example suggests, the reason for this sign change is keeping track of "orientation" for computations. On the other hand, this is necessary as otherwise we cannot obtain some important isomorphisms like

*

*Künneth formula $H^*(M\times N)\cong H^*(M)\otimes H^*(N)$

*uniqueness of determinants
Also, again as the second example suggests, the Koszul sign convention is related with sign of permutations.
