Categorical structures for type theory in univalent foundations

Benedikt Ahrens, Peter Lefanu Lumsdaine, Vladimir Voevodsky

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)
45 Downloads (Pure)


In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in univalent type theory, where the comparisons between them can be given more elementarily than in set-theoretic foundations. Specifically, we construct maps between the various types of structures, and show that assuming the Univalence axiom, some of the comparisons are equivalences. We then analyze how these structures transfer along (weak and strong) equivalences of categories, and, in particular, show how they descend from a category (not assumed univalent/saturated) to its Rezk completion. To this end, we introduce relative universes, generalizing the preceding notions, and study the transfer of such relative universes along suitable structure. We work throughout in (intensional) dependent type theory; some results, but not all, assume the univalence axiom. All the material of this paper has been formalized in Coq, over the UniMath library.
Original languageEnglish
Article number18
Number of pages18
JournalLogical Methods in Computer Science
Issue number3
Publication statusPublished - 11 Sept 2018


  • Categorical Semantics
  • Type Theory
  • Univalence Axiom


Dive into the research topics of 'Categorical structures for type theory in univalent foundations'. Together they form a unique fingerprint.

Cite this