Modern programming language type systems help programmers write correct software, and furthermore helps them write the software they actually intended to write. We show how expressive types can be used to encode dimension and units of measure information, which can be used to avoid dimensional mistakes and guide software construction, and how types can even help to generate code automatically, which eliminates a whole class of bugs.

We provide a categorical framework for models of a type theory that has special types for physical quantities. The types are indexed by the physical dimensions that they involve. Fibrations are used to organize this index structure in the models of the type theory. We develop some informative models of this type theory: firstly, a model based on group actions, which captures invariance under scaling, and secondly, a way of constructing new models using relational parametricity.