A survey article that we have written:
Quantum Gauge Field Theory in Cohesive Homotopy Type Theory
in Proceedings of Quantum Physics and Logic 2012,
Electronic Proceedings in Theoretical Computer Science (2014)
Related material for further reading includes:
A discussion with more emphasis on the differential cohomology hexagon which is axiomatized by cohesive homotopy type theory is in Differential generalized cohomology in Cohesive homotopy type theory.
For a survey of the general picture see at Synthetic Quantum Field Theory.
For a more detailed exposition of the physics involved see the pdf-document at Classical field theory via Cohesive homotopy types.
We implement in the formal language of homotopy type theory a new set of axioms called cohesion. Then we indicate how the resulting cohesive homotopy type theory naturally serves as a formal foundation for central concepts in gauge quantum field theory. This is a brief survey of work by the authors developed in detail elsewhere (Sh, Sc).
Expositions of the general idea are in
For further related references see at differential cohomology in a cohesive topos – References.
Quantum gauge field theory in Cohesive homotopy type theory