Curved $A_{\infty}$-algebras and gauge theory

We propose a general notion of algebraic gauge theory obtained via extracting the main properties of classical gauge theory. Building on a previous work on transferring curved $A_{\infty}$-structures, we show that under certain technical conditions, algebraic gauge theories can be transferred along chain contractions. Specializing to the case of the contraction from differential forms to cochains, we obtain a simplicial gauge theory on the matrix-valued simplicial cochains of a triangulated manifold. In particular, one obtains discrete notions of connection, curvature, gauge transformation and gauge invariant action.