Homotopy Theoretic Aspects of Constructive Type Theory

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Types are weak ω‐groupoids (2008, background): “Whilst attempts to make this link precise have only borne fruit in the past few years [1, 7, 9, 19], the basic idea dates back to an observation made by Hofmann and Streicher in [11].”
Homotopy theoretic models of identity types (2007, background): “L O ] 3 S ep 2 00 7 HOMOTOPY THEORETIC MODELS OF IDENTITY TYPES STEVE AWODEY AND MICHAEL A. WARREN”
The Frobenius condition, right properness, and uniform fibrations (2015, methodology): “ral weak factorization systems in simplicial and cubical sets that have uniform Kan ﬁbrations as the right maps, which are of interest for the interpretation of Martin-Lo¨f’s rules for identity types [2, 15, 34, 4, 30]. These natural weak factorization systems are obtained using Garner’s small objec”
The simplicial model of Univalent Foundations (after Voevodsky) (2012, unknown): “Other methods exist for certain speciﬁc categories ([HS98], [War08]), but are not applicable to the present case.”