What is the internal language of the category Cat of small categories?


I found an [article](http://www.brics.dk/RS/01/27/) by Glynn Winskel and his student Mario Jose Cáccamo about such calculus! However it is limited to a fragment of Cat. I wonder if there have been further work since the PhD thesis of Cáccamo.


<http://www.brics.dk/RS/01/27/>



internal language for the 2-category of small categories
You may be interested in Harper and Licata, *2-Dimensional Directed Dependent Type Theory*, [PDF](http://dlicata.web.wesleyan.edu/pubs/lh102dtt/lh102dtt.pdf). I did some abortive work on this [myself](http://ncatlab.org/michaelshulman/show/internal+logic+of+a+2-category) as well.


