Paolo Capriotti abstract and references

Title: (∞,1)-categories in Homotopy Type Theory

Abstract: We will look at some ideas on how to extend the classical model of(∞,1)-categories given by complete Segal spaces to the setting ofHomotopy Type Theory (HoTT). Since the category Δ is not direct, we are forced to replace bisimplicial sets with semi-simplicial types, which means that categorical identities cannot be immediately recovered from degeneracies.  However, somewhat surprisingly, a notion of completeness can be defined for semi-Segal types, and it seems to be possible to develop a satisfactory theory of (n,1)-categories based on complete semi-Segal types internally in HoTT.

References:

Univalence for inverse diagrams and homotopy canonicity

https://arxiv.org/abs/1203.3253

Univalent categories and the Rezk completion

http://arxiv.org/abs/1303.0584

Quasi-unital ∞-Categories

http://arxiv.org/pdf/1210.0212.pdf

Concrete Categories in Homotopy Type Theory

http://arxiv.org/abs/1311.1852

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