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The Yang–Mills field is the gauge field of Yang-Mills theory.
It is modeled by a cocycle $\hat F \in \mathbf{H}(X, \mathbf{B}U(n)_{conn})$ in differential nonabelian cohomology. Here $\mathbf{B} U(n)_{conn}$ is the moduli stack of $U(n)$-principal connections, the stackification of the groupoid of Lie-algebra valued forms, regarded as a groupoid internal to smooth spaces.
This is usually represented by a vector bundle with connection.
As a nonabelian Čech cocycle the Yang-Mills field on a space $X$ is represented by
a cover $\{U_i \to X\}$
a collection of $Lie(U(n))$-valued 1-forms $(A_i \in \Omega^1(U_i, Lie(U(n))))$;
a collection of $U(n)$-valued smooth functions $(g_{i j} \in C^\infty(U_{i j}, U(n)))$;
such that on double overlaps
and such that on triple overlaps
For $U(n) = U(1)$ this is the electromagnetic field.
For $U(n) = SU(2) \times U(1)$ this is the “electroweak field”;
For $U(n) = SU(3)$ this is the strong nuclear force field.