  In this article we construct and discuss a new rigorous geometric formalism
for gauge field theories. The basis of our work is the notion of the Tulczyjew
triple, a geometric structure which successfully solved numerous problems in
mathematical description of mechanics and classical field theory. In
particular, we construct a Tulczyjew triple for gauge field theories and reduce
it for systems that depend only on the value of a connection and curvature
instead of the entire first jet of the gauge field. We also introduce new
geometric structures such as the vector-affine product of bundles and analyse
the connection bundle from a new perspective. Finally, we apply the derived
formalism to Yang-Mills theory.
