group cohomology - Why is the standard definition of cocycle the one that _always_ comes up??

Late to the party as usual, but: the goal of this answer is to convince you that the standard convention for $2$-cocycles is so natural that you should consider it perverse to consider any other convention, modulo "applying a canonical involution to everything," as you say. To keep things simple let's only deal with trivial action on coefficients. The motivating question is the following:

What does it mean for a group $G$ to act on a category $C$?

For starters we should attach to each element of $G$ a functor $F(g) : C to C$. Next we could require that $F(g) circ F(h) = F(gh)$, but we should really weaken equalities of functors to natural isomorphisms whenever possible. Hence we should attach to each pair of elements of $G$ a natural isomorphism

$$eta(g, h) : F(g) circ F(h) to F(gh).$$

This is the point at which we pick a convention for how we're going to represent $2$-cocycles. Instead of talking about $eta(g, h)$ we could talk about its inverse; which we choose corresponds to whether we prefer to talk about lax monoidal or oplax monoidal functors, since what we're going to end up writing down is a lax monoidal resp. an oplax monoidal functor from $G$ (regarded as a discrete monoidal category) to $text{Aut}(C)$ (regarded as a monoidal category under composition).

In any case, let's stick to the above choice (the lax one). Then the isomorphisms $eta(g, h)$ should satisfy some coherence conditions, the important one being the "associativity" condition that the two obvious ways of going from $F(g_1) circ F(g_2) circ F(g_3)$ to $F(g_1 g_2 g_3)$ should agree.

Now let's assume that in addition all of the functors $F(g)$ are the identity functor $text{id}_C : C to C$. Then the only remaining data in a group action is a collection of natural automorphisms

$$eta(g, h) : text{id}_C to text{id}_C$$

of the identity functor. For any category $C$, the natural automorphisms of the identity functor naturally form an abelian (by the Eckmann-Hilton argument) group which here I'll call its center $Z(C)$ (but this notation is also used for the commutative monoid of natural endomorphisms of the identity). So we get a function

$$eta : G times G to Z(C).$$

The important coherence condition I mentioned above now reduces (again by the Eckmann-Hilton argument) to the condition that for any $g_1, g_2, g_3 in G$ we have

$$eta(g_1, g_2) eta(g_1 g_2, g_3) = eta(g_2, g_3) eta(g_1, g_2 g_3)$$

which is precisely the standard cocycle condition. (Coboundaries come in when you ask what it means for two group actions to be equivalent; I'm going to ignore this.)

The only reason this condition, which recall is in general just the statement that the two obvious ways of going from $F(g_1) circ F(g_2) circ F(g_3)$ to $F(g_1 g_2 g_3)$ should agree, could ever have looked anything other than completely natural is that it's a degenerate special case where the sources and targets of the various maps involved have been obscured because they are identical. In particular, of course I could have instead chosen to think about the natural isomorphisms

$$eta(g, g^{-1} h) : F(g) circ F(g^{-1} h) to F(h)$$

(which corresponds to your $f(1, g, h)$), but now

• it's no longer at all obvious how to state the associativity condition succinctly, and


• this requires that I make explicit use of the fact that $G$ is a group.

The discussion up til now in fact gives a perfectly reasonable definition for what it means for a monoid to act on a category. (If I want to weaken "natural isomorphism" to "natural transformation," though, I get two genuinely different possibilities depending on whether I pick lax or oplax monoidal functors.)

Reflecting on associativity suggests that, for a more "unbiased" point of view, we should consider families of natural isomorphisms

$$eta(g_1, g_2, dots g_n) : F(g_1) circ F(g_2) circ dots circ F(g_n) to F(g_1 g_2 dots g_n)$$

and then impose a "generalized associativity" condition that every way of composing them to get a natural isomorphism with the same source and target as $eta(g_1, g_2, dots g_n)$ should give $eta(g_1, g_2, dots g_n)$. Another way to say this is that the cocycle condition (in the $F(g) = text{id}_C$ special case, at least) should really be written

$$eta(g_1, g_2, g_3) = eta(g_1, g_2) eta(g_1 g_2, g_3) = eta(g_2, g_3) eta(g_1, g_2 g_3).$$

This is in the same way that we can consider a monoid operation to be a family $m(g_1, g_2, dots g_n) = g_1 g_2 dots g_n$ of operations satisfying a generalized associativity condition, and in particular satisfying

$$m(g_1, g_2, g_3) = m(m(g_1, g_2), g_3) = m(g_1, m(g_2, g_3)).$$

Namely, by "associativity" we usually mean that the middle expression equals the right, but really the reason that the middle expression equals the right is that they both equal the left.

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