Categorical Structures for Type Theory in Univalent Foundations - Inria - Institut national de recherche en sciences et technologies du numérique Accéder directement au contenu
Communication Dans Un Congrès Année : 2017

Categorical Structures for Type Theory in Univalent Foundations

Résumé

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 the setting of univalent foundations, where the relationships between them can be stated more transparently. Specifically, we construct maps between the different structures and show that these maps are equivalences under suitable assumptions. 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.
Fichier principal
Vignette du fichier
LIPIcs-CSL-2017-8.pdf (543.9 Ko) Télécharger le fichier
Origine : Fichiers éditeurs autorisés sur une archive ouverte
Loading...

Dates et versions

hal-01579271 , version 1 (30-08-2017)

Identifiants

Citer

Benedikt Ahrens, Peter Lefanu Lumsdaine, Vladimir Voevodsky. Categorical Structures for Type Theory in Univalent Foundations . 26th EACSL Annual Conference on Computer Science Logic (CSL 2017), Aug 2017, Stockholm, Sweden. pp.8:1 - 8:16, ⟨10.4230/LIPIcs.CSL.2017.8⟩. ⟨hal-01579271⟩
301 Consultations
250 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More