Abstract

I will run through the basics of how dependent types are modelled inside a category, including how to construct dependent sums and products, and how to represent substitution. I will then discuss the coherence issues relating to substitution and how these can be resolved using a category with families (CwF) or similar construction (e.g. comprehension categories, categories with attributes etc).

Covering:

• Dependent types as fibrations
• Dependent sum/product as adjoints to the pullback functor
• Substitution via pullback and its problems
• Fixing those problems with categories with families (CwFs)

Prerequisites:

• Basic category theory (functors and pullbacks)
• Basic knowledge of dependent types