We study different formalizations of finite sets in homotopy type theory to obtain a general definition that exhibits both the computational facilities and the proof principles expected from finite sets. We use higher inductive types to define the type K(A) of łfinite sets over type Až à la Kuratowski without assuming that A has decidable equality. We show how to define basic functions and prove basic properties after which we give two applications of our definition.On the foundational side, we use K to define the notions of łKuratowski-finite typež and łKuratowski-finite subobjectž, which we contrast with established notions, e.g., Bishop-finite types and enumerated types. We argue that Kuratowski-finiteness is the most general and flexible one of those and we define the usual operations on finite types and subobjects.From the computational perspective, we show how to use K(A) for an abstract interface for well-known finite set implementations such as tree-and list-like data structures. This implies that a function defined on a concrete finite sets implementation can be obtained from a function defined on the abstract finite sets K(A) and that correctness properties are inherited. Hence, HoTT is the ideal setting for data refinement. Beside this, we define bounded quantification, which lifts a decidable property on A to one on K(A).

We study different formalizations of finite sets in homotopy type theory to obtain a general definition that exhibits both the computational facilities and the proof principles expected from finite sets. We use higher inductive types to define the type K(A) of łfinite sets over type Až à la Kuratowski without assuming that A has decidable equality. We show how to define basic functions and prove basic properties after which we give two applications of our definition.On the foundational side, we use K to define the notions of łKuratowski-finite typež and łKuratowski-finite subobjectž, which we contrast with established notions, e.g., Bishop-finite types and enumerated types. We argue that Kuratowski-finiteness is the most general and flexible one of those and we define the usual operations on finite types and subobjects.From the computational perspective, we show how to use K(A) for an abstract interface for well-known finite set implementations such as tree-and list-like data structures. This implies that a function defined on a concrete finite sets implementation can be obtained from a function defined on the abstract finite sets K(A) and that correctness properties are inherited. Hence, HoTT is the ideal setting for data refinement. Beside this, we define bounded quantification, which lifts a decidable property on A to one on K(A).

We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples of univalent bicategories in a modular fashion, we develop displayed bicategories, an analog of displayed 1-categories introduced by Ahrens and Lumsdaine. We demonstrate the applicability of this notion and prove that several bicategories of interest are univalent. Among these are the bicategory of univalent categories with families and the bicategory of pseudofunctors between univalent bicategories. Furthermore, we show that every bicategory with univalent hom-categories is weakly equivalent to a univalent bicategory. All of our work is formalized in Coq as part of the UniMath library of univalent mathematics.

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In this paper, we show that all finitary 1-truncated higher inductive types (HITs) can be constructed from the groupoid quotient. We start by defining internally a notion of signatures for HITs, and for each signature, we construct a bicategory of algebras in 1-types and in groupoids. We continue by proving initial algebra semantics for our signatures. After that, we show that the groupoid quotient induces a biadjunction between the bicategories of algebras in 1-types and in groupoids. We finish by constructing a biinitial object in the bicategory of algebras in groupoids. From all this, we conclude that all finitary 1-truncated HITs can be constructed from the groupoid quotient. All the results are formalized over the UniMath library of univalent mathematics in Coq.