Abstract. Higher inductive types (HITs) in Homotopy TypeTheory allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types, and allow to define types with non-trivial higher equality types, such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define types satisfying uniqueness of equality proofs, such as the Cauchy reals, the partiality monad, and the well-typed syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductive-inductive definitions. We call those HITs quotient inductive-inductive types (QIITs). Although there has been recent progress on a general theory of HITs, there is not yet a theoretical foundation for the combination of equality constructors and induction-induction, despite many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initial-algebra semantics. We further derive a section induction principle, stating that every algebra morphism into the algebra in question has a section, which is close to the intuitively expected elimination rules.

Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the loop space of the suspension of A + 1. Equiv-

Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (Ahrens et al., 2015) solves this by considering only truncated types, roughly corresponding to an ordinary category. This fails to capture many naturally occurring structures, stemming from the fact that the naturally occurring structures in homotopy type theory are not ordinary, but rather higher categories.Out of the large variety of approaches to higher category theory that mathematicians have proposed, we believe that, for type theory, the simplicial strategy is best suited. Work by Lurie (2009a) and Harpaz (2015) motivates the following definition. Given the first (n+3) levels of a semisimplicial type S, we can equip S with three properties: first, contractibility of the types of certain horn fillers; second, a completeness property; and third, a truncation condition. We call this a complete semi-Segal n-type. This is very similar to an earlier suggestion by Schreiber (2012).The definition of a univalent (1-) category in (Ahrens et al., 2015) can easily be extended or restricted to the definition of a univalent n-category (more precisely, (n, 1)-category) for n ∈ {0, 1, 2}, and we show that the type of complete semi-Segal n-types is equivalent to the type of univalent n-categories in these cases. Thus, we believe that the notion of a complete semi-Segal n-type can be taken as the definition of a univalent n-category.We provide a formalisation in the proof assistant Agda using a completely explicit representation of semi-simplicial types for levels up to 4. This is the full version of the paper, containing all proofs. An abridged version titled Univalent Higher Categories via Complete Semi-Segal Types will be presented at the 45th ACM SIGPLAN Symposium on Principles of Programming Languages (POPL 2018).

We define and develop two-level type theory, a version of Martin-Löf type theory which is able to combine two type theories. In our case of interest, the first of these two theories is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The second is a traditional form of type theory validating uniqueness of identity proofs (UIP) and may be understood as internalised meta-theory of the first.We show how two-level type theory can be used to address some of the open issues of HoTT, including the construction of "infinite structures" such as a universe of semi-simplicial types, and the internal definition of higher categorical structures.The idea of two-level type theory is heavily inspired by Voevodsky's Homotopy Type System (HTS). Two-level type theory can be thought of as a version of HTS without equality reflection. We show that the lack of equality reflection does not hinder the development of the ideas that HTS was designed for.Some of the results of this paper have been formalised in the proof assistant Lean, in order to demonstrate how two-level type theory, despite being an extension of the underlying type theory of current proof assistants, can be encoded in existing systems in a way that is both practical and easy to implement.

The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed.We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions.To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman [26]), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman [18]).