In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces and maps of graphs embedded in the sphere, in homotopy type theory (HoTT). This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for HoTT.

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.

We develop the Scott model of the programming language PCF in univalent type theory. Moreover, we work constructively and predicatively. To account for the non-termination in PCF, we use the lifting monad (also known as the partial map classifier monad) from topos theory, which has been extended to univalent type theory by Escardó and Knapp. Our results show that lifting is a viable approach to partiality in univalent type theory. Moreover, we show that the Scott model can be constructed in a predicative and constructive setting. Other approaches to partiality either require some form of choice or quotient inductive-inductive types. We show that one can do without these extensions.

We investigate the injective types and the algebraically injective types in univalent mathematics, both in the absence and in the presence of propositional resizing. Injectivity is defined by the surjectivity of the restriction map along any embedding, and algebraic injectivity is defined by a given section of the restriction map along any embedding. Under propositional resizing axioms, the main results are easy to state: (1) Injectivity is equivalent to the propositional truncation of algebraic injectivity. (2) The algebraically injective types are precisely the retracts of exponential powers of universes. (2a) The algebraically injective sets are precisely the retracts of powersets. (2b) The algebraically injective (n+1)-types are precisely the retracts of exponential powers of universes of n-types. (3) The algebraically injective types are also precisely the retracts of algebras of the partial-map classifier. From (2) it follows that any universe is embedded as a retract of any larger universe. In the absence of propositional resizing, we have similar results that have subtler statements which need to keep track of universe levels rather explicitly, and are applied to get the results that require resizing.