Constructing Higher Inductive Types as Groupoid Quotients

In this paper, we study finitary 1-truncated higher inductive types (HITs) in homotopy type theory. We start by showing that all these can be constructed from the groupoid quotient. define an internal notion of signatures for HITs, and each signature, construct a bicategory algebras 1-types groupoids. continue proving initial algebra semantics our signatures. After that, show quotient induces biadjunction between bicategories Then biinitial object groupoids, which gives desired algebra. From this, conclude HITs present several examples are definable using signature. particular, signature rise to HIT corresponding freely generated algebraic structure over it. also development universal 1-types. has PIE limits, i.e. products, inserters equifiers, prove version first isomorphism theorem Finally, give alternative characterization foundamental groups some exploiting construction via All results formalized UniMath library univalent mathematics Coq.