Projects in Previous Years

1st Semester 2003/04: Categories of Interpretations


Relative interpretability is one of the most common ways of of comparing theories. E.g. `the Poincaré model' is a relative interpretation of two-dimensional hyperbolic geometry in two dimensional Eucidean geometry.

Surprisingly almost nothing is known about the category of interpretations. A moment of thought shows that there is in fact not one category of interpretations but that there are several depending on the answer to the question: when are two interpretations the same?

The project is to do some groundwork to settle some absolutely basic questions:

  • Comparison of interpretations with or without parameters.
  • Comparison of interpretations with or without relativization and with or without preservation of identity.
  • Under what circumstances do sums exist? (I have a full answer for products.)
  • What about equalizers and co-equalizers?
  • Separating examples between the various categories.
  • Classification of famous interpretations. E.g. is `the PoincarĂ© model' a monomorphism?
  • Should further structure be added to the categories to make them a fruitful tool to formulate classical results?

I will provide an introduction to the basic notions required to work on the project. Some knowledge of elementary category theory is helpful, but not strictly necessary.

Location: The project will take place in Utrecht.