Let $K$ be a field, $G=Gal(\overline{K}/K)$ be its absolute Galois group, and let $M$ be a discrete $G$-module.
We define Galois cohomology as follows.
$H^1(G,M)\stackrel{\mathrm{def}}{=}Z^1(G,M)/B^1(G,M)$.
Here, $Z^1(G,M)=\{f:G\to M :\text{continuous} \mid f(\sigma \tau)=f(\sigma)+\sigma f(\tau), \forall \sigma,\tau \in G\}$, $B^1(G,M)\stackrel{\mathrm{def}}{=} \{f:G\to M:\text{continuous}\mid \exists m \in M, \forall \sigma \in G, f(\sigma)=\sigma m-m \}$.
The definition of $Z^1(G,M)$ is natural for me. It is a group of crossed homomorphisms.
On the other hand, the definition of $B^1(G,M)$ is not natural for me. $B^1(G,M)$ is collection of continuous maps $G\to M$ which are written by simple cocycles $\sigma m-m$. But where does $\sigma m -m$ come from? Why do we think equivalent relation by $B^1(G,M)$?
Of course, definition by more abstract way gives this is an image of a certain boundary map.
Based on these thoughts on $B^1(G,M)$, I would like to hear your thoughts on the definition on $B^1(G,M)$.