We give a brief overview of the special issue of MSCS “Advances in Homotopy Type Theory.”

We take another look at the construction by Hofmann and Streicher of a universe $(U,{\mathcal{E}l})$ for the interpretation of Martin-Löf type theory in a presheaf category $[{{{\mathbb{C}}}^{\textrm{op}}},\textsf{Set}]$. It turns out that $(U,{\mathcal{E}l})$ can be described as the nerve of the classifier $\dot{{\textsf{Set}}}^{\textsf{op}} \rightarrow{{\textsf{Set}}}^{\textsf{op}}$ for discrete fibrations in $\textsf{Cat}$, where the nerve functor is right adjoint to the so-called “Grothendieck construction” taking a presheaf $P :{{{\mathbb{C}}}^{\textrm{op}}}\rightarrow{\textsf{Set}}$ to its category of elements $\int _{\mathbb{C}} P$. We also consider change of base for such universes, as well as universes of structured families, such as fibrations.

We develop realizability models of intensional type theory, based on groupoids, wherein realizers themselves carry non-trivial (non-discrete) homotopical structure. In the spirit of realizability, this is intended to formalize a homotopical BHK interpretation, whereby evidence for an identification is a path. Specifically, we study partitioned groupoidal assemblies. Categories of such are parameterized by “realizer categories” (instead of the usual partial combinatory algebras) that come equipped with an interval qua internal cogroupoid. The interval furnishes a notion of homotopy as well as a fundamental groupoid construction. Objects in a base groupoid are realized by points in the fundamental groupoid of some object from the realizer category; isomorphisms in the base groupoid are realized by paths in said fundamental groupoid. The main result is that, under mild conditions on the realizer category, the ensuing category of partitioned groupoidal assemblies models intensional (1-truncated) type theory without function extensionality. Moreover, when the underlying realizer category is “untyped,” there exists an impredicative universe of 1-types (the modest fibrations). This is a groupoidal analog of the traditional situation.

When working in homotopy type theory and univalent foundations, the traditional role of the category of sets, $\mathcal{Set}$, is replaced by the category $\mathcal{hSet}$ of homotopy sets (h-sets); types with h-propositional identity types. Many of the properties of $\mathcal{Set}$ hold for $\mathcal{hSet}$ ((co)completeness, exactness, local cartesian closure, etc.). Notably, however, the univalence axiom implies that $\mathsf{Ob}\,\mathcal{hSet}$ is not itself an h-set, but an h-groupoid. This is expected in univalent foundations, but it is sometimes useful to also have a stricter universe of sets, for example, when constructing internal models of type theory. In this work, we equip the type of iterative sets $\mathsf{V}^0$, due to Gylterud ((2018). The Journal of Symbolic Logic 83 (3) 1132–1146) as a refinement of the pioneering work of Aczel ((1978). Logic Colloquium’77, Studies in Logic and the Foundations of Mathematics, vol. 96, Elsevier, 55–66.) on universes of sets in type theory, with the structure of a Tarski universe and show that it satisfies many of the good properties of h-sets. In particular, we organize $\mathsf{V}^0$ into a (non-univalent strict) category and prove that it is locally cartesian closed. This enables us to organize it into a category with families with the structure necessary to model extensional type theory internally in HoTT/UF. We do this in a rather minimal univalent type theory with W-types, in particular we do not rely on any HITs, or other complex extensions of type theory. Furthermore, the construction of $\mathsf{V}^0$ and the model is fully constructive and predicative, while still being very convenient to work with as the decoding from $\mathsf{V}^0$ into h-sets commutes definitionally for all type constructors. Almost all of the paper has been formalized in $\texttt{Agda}$ using the $\texttt{agda}$-$\texttt{unimath}$ library of univalent mathematics.

This paper explain how the geometric notions of local contractibility and properness are related to the $\Sigma$-types and $\Pi$-types constructors of dependent type theory. We shall see how every Grothendieck fibration comes canonically with such a pair of notions—called smooth and proper maps—and how this recovers the previous examples and many more. This paper uses category theory to reveal a common structure between geometry and logic, with the hope that the parallel will be beneficial to both fields. The style is mostly expository, and the main results are proved in external references.

In homotopy type theory, few constructions have proved as troublesome as the smash product. While its definition is just as direct as in classical mathematics, one quickly realises that in order to define and reason about functions over iterations of it, one has to verify an exponentially growing number of coherences. This has led to crucial results concerning smash products remaining open. One particularly important such result is the fact that the smash product forms a (1-coherent) symmetric monoidal product on the universe of pointed types. This fact was used, without a complete proof, by, for example, Brunerie ((2016) PhD thesis, Université Nice Sophia Antipolis) to construct the cup product on integral cohomology and is, more generally, a fundamental result in traditional algebraic topology. In this paper, we salvage the situation by introducing a simple informal heuristic for reasoning about functions defined over iterated smash products. We then use the heuristic to verify, for example, the hexagon and pentagon identities, thereby obtaining a proof of symmetric monoidality. We also provide a formal statement of the heuristic in terms of an induction principle concerning the construction of homotopies of functions defined over iterated smash products. The key results presented here have been formalised in the proof assistant Cubical Agda.

This is a foundation for algebraic geometry, developed internal to the Zariski topos, building on the work of Kock and Blechschmidt (Kock (2006) [I.12], Blechschmidt (2017)). The Zariski topos consists of sheaves on the site opposite to the category of finitely presented algebras over a fixed ring, with the Zariski topology, that is, generating covers are given by localization maps for finitely many elements $f_1,\dots, f_n$ that generate the ideal $(1)=A\subseteq A$. We use homotopy-type theory together with three axioms as the internal language of a (higher) Zariski topos. One of our main contributions is the use of higher types – in the homotopical sense – to define and reason about cohomology. Actually computing cohomology groups seems to need a principle along the lines of our “Zariski local choice” axiom, which we justify as well as the other axioms using a cubical model of homotopy-type theory.