give an idea of what category theory is and how it can be successfully applied in mathematics and the mathematical sciences by means of example. The example is a notion from mathematical logic formalizing the intuitive concept of "property". The new category-theoretical
definition of this notion can physically be interpreted as a measurement. Unraveling this definition in particular categories can be regarded as defining the concept of "property" in different context, e.g. in classical, probabilistic
and quantum physics, and in each case this recovers familiar things.
Up to some differences in the technical details, all of this is work of Bart Jacobs. (See arXiv:1205.3940)