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Categorical models of dependent types

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If you have a question about this talk, please contact Ian Orton.

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).


  • 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)


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

This talk is part of the Logic & Semantics for Dummies series.

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