I have a question regarding the mathematical description of Yang-Mills theories. In classical Yang-Mills theory, the Yang-Mills field (or field strength in physical language) is given by the curvature $F$ of the corresponding principal fibre bundle, which means that it is a Lie-algebra valued 2-form.
From mathematical Quantum Field Theory, I know that quantized fields are normally described as operator-valued distributions...
How does this two concepts combine in the case of quantized Yang-Mills Theory?