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Higher categories from type theories

This thesis continues the programme of providing a higher-categorical analysis of the treatment of equality in Martin-Lof dependent type theory. 
In particular, we construct for various type theories a classifying weak ω-category, with objects and 1-cells as in the standard classifying category, and higher cells being open terms of identity types between these. Weak ω-category structures (in the sense of Batanin/Leinster) on these are given by operads of syntactically definable composition laws.