On categorical models of dependent types via locally Cartesian closed categories:
Daniel Gratzer, G. Alex Kavvos, Andreas Nuyts, Lars Birkedal: Multimodal Dependent Type Theory, Logical Methods in Computer Science 17 3 (2021) lmcs:7713 [arXiv:2011.15021, doi:10.46298/lmcs-17(3:11)2021]
Daniel Gratzer, Implementing Modal Dependent Type Theory, talk at ICFP 19 (slides pdf)
Daniel Gratzer, Syntax and semantics of modal type theory, PhD Aarhus (2023) [pdf]
On XTT:
Jonathan Sterling, Carlo Angiuli, Daniel Gratzer, Cubical syntax for reflection-free extensional equality. In Herman Geuvers (ed.), 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019), volume 131 of Leibniz International Proceedings in Informatics (LIPIcs), pages 31:1-31:25. (arXiv:1904.08562, doi:10.4230/LIPIcs.FCSD.2019.31)
Jonathan Sterling, Carlo Angiuli, Daniel Gratzer, A Cubical Language for Bishop Sets, Logical Methods in Computer Science, 18 (1), 2022. (arXiv:2003.01491).
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