For $c \in \Gamma^\bullet(X)$ a cocycle representing a gauge field in gauge theory, its image $F(c) \in \Omega^\bullet(X)$ is the field strength of the gauge field. If we think of this cocycle as being (a generalization of) a connection on a bundle, this is essentially the curvature of that connection.

Often gauge fields are named after their field strength. For instance the field strength of the electromagnetic field is the $2$-form $F \in \Omega^2(X)$ whose components are the electric and the magnetic fields.