arXiv subjects

Daniel Gratzer

Publications and source records attributed to Daniel Gratzer.

18 recordsLinked to original sources

arXiv2026

The $\infty$-category of $\infty$-categories in simplicial type theory

Simplicial type theory (STT) was introduced by Riehl and Shulman to leverage homotopy type theory to prove results about $(\infty,1)$-categories. Initial work on simplicial type theory focused on "formal" arguments in higher category theory and, in particular, no non-trivial examples of $\infty$-category theory were constructible within STT. More recent work has changed this state of affairs by applying techniques developed initial for cubical type theory to construct the $\infty$-category of spaces. We complete this process by constructing the $\infty$-category of $\infty$-categories, recovering one of the main foundational results of $\infty$-category theory (straightening--unstraightening) purely type-theoretically. We also show how this construction enables new examples of the directed version of the structure identity principle, the structure homomorphism principle.

cs.LO↗

arXiv2025

The Yoneda embedding in simplicial type theory

Riehl and Shulman introduced simplicial type theory (STT), a variant of homotopy type theory which aimed to study not just homotopy theory, but its fusion with category theory: $(\infty,1)$-category theory. While notoriously technical, manipulating $\infty$-categories in simplicial type theory is often easier than working with ordinary categories, with the type theory handling infinite stacks of coherences in the background. We capitalize on recent work by Gratzer et al. defining the $(\infty,1)$-category of $\infty$-groupoids in STT to define presheaf categories within STT and systematically develop their theory. In particular, we construct the Yoneda embedding, prove the universal property of presheaf categories, refine the theory of adjunctions in STT, introduce the theory of Kan extensions, and prove Quillen's Theorem A.

cs.LO↗

arXiv2024

Directed univalence in simplicial homotopy type theory

Simplicial type theory extends homotopy type theory with a directed path type which internalizes the notion of a homomorphism within a type. This concept has significant applications both within mathematics -- where it allows for synthetic (higher) category theory -- and programming languages -- where it leads to a directed version of the structure identity principle. In this work, we construct the first types in simplicial type theory with non-trivial homomorphisms. We extend simplicial type theory with modalities and new reasoning principles to obtain triangulated type theory in order to construct the universe of discrete types $\mathcal{S}$. We prove that homomorphisms in this type correspond to ordinary functions of types i.e., that $\mathcal{S}$ is directed univalent. The construction of $\mathcal{S}$ is foundational for both of the aforementioned applications of simplicial type theory. We are able to define several crucial examples of categories and to recover important results from category theory. Using $\mathcal{S}$, we are also able to define various types whose usage is guaranteed to be functorial. These provide the first complete examples of the proposed directed structure identity principle.

cs.LO↗

arXiv2024

The Category of Iterative Sets in Homotopy Type Theory and Univalent Foundations

When working in Homotopy Type Theory and Univalent Foundations, the traditional role of the category of sets, Set, is replaced by the category hSet of homotopy sets (h-sets); types with h-propositional identity types. Many of the properties of Set hold for hSet ((co)completeness, exactness, local cartesian closure, etc.). Notably, however, the univalence axiom implies that Ob(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 V0, due to Gylterud (2018) as a refinement of the pioneering work of Aczel (1978) 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 V0 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 V0 and the model is fully constructive and predicative, while still being very convenient to work with as the decoding from V0 into h-sets commutes definitionally for all type constructors. Almost all of the paper has been formalized in Agda using the agda-unimath library of univalent mathematics.

cs.LO↗

arXiv2023

Towards univalent reference types

We develop a denotational semantics for general reference types in an impredicative version of guarded homotopy type theory, an adaptation of synthetic guarded domain theory to Voevodsky's univalent foundations. We observe for the first time the profound impact of univalence on the denotational semantics of mutable state. Univalence automatically ensures that all computations are invariant under symmetries of the heap -- a bountiful source of program equivalences. In particular, even the most simplistic univalent model enjoys many new equations that do not hold when the same constructions are carried out in the universes of traditional set-level (extensional) type theory.

cs.LO↗

arXiv2023

Normalization for multimodal type theory

We prove normalization for MTT, a general multimodal dependent type theory capable of expressing modal type theories for guarded recursion, internalized parametricity, and various other prototypical modal situations. We prove that deciding type checking and conversion in MTT can be reduced to deciding the equality of modalities in the underlying modal situation, immediately yielding a type checking algorithm for all instantiations of MTT in the literature. This proof uses a generalization of synthetic Tait computability -- an abstract approach to gluing proofs -- to account for modalities. This extension is based on MTT itself, so that this proof also constitutes a significant case study of MTT.

cs.LO↗

arXiv2022

Under Lock and Key: A Proof System for a Multimodal Logic

We present a proof system for a multimodal logic, based on our previous work on a multimodal Martin-Loef type theory. The specification of modes, modalities, and implications between them is given as a mode theory, i.e. a small 2-category. The logic is extended to a lambda calculus, establishing a Curry-Howard correspondence.

cs.LO↗

arXiv2022

Controlling unfolding in type theory

We present a new way to control the unfolding of definitions in dependent type theory. Traditionally, proof assistants require users to fix whether each definition will or will not be unfolded in the remainder of a development; unfolding definitions is often necessary in order to reason about them, but an excess of unfolding can result in brittle proofs and intractably large proof goals. In our system, definitions are by default not unfolded, but users can selectively unfold them in a local manner. We justify our mechanism by means of elaboration to a core theory with extension types -- a connective first introduced in the context of homotopy type theory -- and by establishing a normalization theorem for our core calculus. We have implemented controlled unfolding in the cooltt proof assistant, inspiring an independent implementation in Agda.

cs.LO↗

arXiv2022

Denotational semantics of general store and polymorphism

We contribute the first denotational semantics of polymorphic dependent type theory extended by an equational theory for general (higher-order) reference types and recursive types, based on a combination of guarded recursion and impredicative polymorphism; because our model is based on recursively defined semantic worlds, it is compatible with polymorphism and relational reasoning about stateful abstract datatypes. We then extend our language with modal constructs for proof-relevant relational reasoning based on the "logical relations as types" principle, in which equivalences between imperative abstract datatypes can be established synthetically. What is new in relation to prior typed denotational models of higher-order store is that our Kripke worlds need not be syntactically definable, and are thus compatible with relational reasoning in the heap. Our work combines recent advances in the operational semantics of state with the purely denotational viewpoint of synthetic guarded domain theory.

cs.PL↗

arXiv2022

Unifying cubical and multimodal type theory

In this paper we combine the principled approach to modalities from multimodal type theory (MTT) with the computationally well-behaved realization of identity types from cubical type theory (CTT). The result -- cubical modal type theory (Cubical MTT) -- has the desirable features of both systems. In fact, the whole is more than the sum of its parts: Cubical MTT validates desirable extensionality principles for modalities that MTT only supported through ad hoc means. We investigate the semantics of Cubical MTT and provide an axiomatic approach to producing models of Cubical MTT based on the internal language of topoi and use it to construct presheaf models. Finally, we demonstrate the practicality and utility of this axiomatic approach to models by constructing a model of (cubical) guarded recursion in a cubical version of the topos of trees. We then use this model to justify an axiomatization of L\"ob induction and thereby use Cubical MTT to smoothly reason about guarded recursion.

cs.LO↗

arXiv2022

Strict universes for Grothendieck topoi

Hofmann and Streicher famously showed how to lift Grothendieck universes into presheaf topoi, and Streicher has extended their result to the case of sheaf topoi by sheafification. In parallel, van den Berg and Moerdijk have shown in the context of algebraic set theory that similar constructions continue to apply even in weaker metatheories. Unfortunately, sheafification seems not to preserve an important realignment property enjoyed by the presheaf universes that plays a critical role in models of univalent type theory as well as synthetic Tait computability, a recent technique to establish syntactic properties of type theories and programming languages. In the context of multiple universes, the realignment property also implies a coherent choice of codes for connectives at each universe level, thereby interpreting the cumulativity laws present in popular formulations of Martin-L\"of type theory. We observe that a slight adjustment to an argument of Shulman constructs a cumulative universe hierarchy satisfying the realignment property at every level in any Grothendieck topos. Hence one has direct-style interpretations of Martin-L\"of type theory with cumulative universes into all Grothendieck topoi. A further implication is to extend the reach of recent synthetic methods in the semantics of cubical type theory and the syntactic metatheory of type theory and programming languages to all Grothendieck topoi.

math.CT↗

arXiv2022

The directed plump ordering

Based on Taylor's hereditarily directed plump ordinals, we define the directed plump ordering on W-types in Martin-L\"of type theory. This ordering is similar to the plump ordering but comes equipped with non-empty finite joins in addition to the usual properties of the plump ordering.

cs.LO↗

arXiv2022

An inductive-recursive universe generic for small families

We show that it is possible to construct a universe in all Grothendieck topoi with injective codes a la Pujet and Tabareau which is nonetheless generic for small families. As a trivial consequence, we show that their observational type theory admits interpretations in Grothendieck topoi suitable for use as internal languages.

cs.LO↗

arXiv2021

Normalization for multimodal type theory

We consider the conversion problem for multimodal type theory (MTT) by characterizing the normal forms of the type theory and proving normalization. Normalization follows from a novel adaptation of Sterling's Synthetic Tait Computability which generalizes the framework to accommodate a type theory with modalities and multiple modes. As a corollary of our main result, we reduce the conversion problem of MTT to the conversion problem of its mode theory and show the injectivity of type constructors. Finally, we conclude that MTT enjoys decidable type-checking when instantiated with a decidable mode theory.

cs.LO↗

arXiv2020

Syntactic categories for dependent type theory: sketching and adequacy

We argue that locally Cartesian closed categories form a suitable doctrine for defining dependent type theories, including non-extensional ones. Using the theory of sketches, one may define syntactic categories for type theories in a style that resembles the use of Martin-L\"of's Logical Framework, following the "judgments as types" principle. The concentration of type theories into their locally Cartesian closed categories of judgments is particularly convenient for proving syntactic metatheorems by semantic means (canonicity, normalization, etc.). Perhaps surprisingly, the notion of a context plays no role in the definitions of type theories in this sense, but the structure of a class of display maps can be imposed on a theory post facto wherever needed, as advocated by the Edinburgh school and realized by the %worlds declarations of the Twelf proof assistant. Uemura has proposed representable map categories together with a stratified logical framework for similar purposes. The stratification in Uemura's framework restricts the use of dependent products to be strictly positive, in contrast to the tradition of Martin-L\"of's logical framework and Schroeder-Heister's analysis of higher-level deductions. We prove a semantic adequacy result for locally Cartesian closed categories relative to Uemura's representable map categories: if a theory is definable in the framework of Uemura, the locally Cartesian closed category that it generates is a conservative (fully faithful) extension of its syntactic representable map category. On this basis, we argue for the use of locally Cartesian closed categories as a simpler alternative to Uemura's representable map categories.

cs.LO↗

arXiv2020

Multimodal Dependent Type Theory

We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We show that different choices of mode theory allow us to use the same type theory to compute and reason in many modal situations, including guarded recursion, axiomatic cohesion, and parametric quantification. We reproduce examples from prior work in guarded recursion and axiomatic cohesion, thereby demonstrating that MTT constitutes a simple and usable syntax whose instantiations intuitively correspond to previous handcrafted modal type theories. In some cases, instantiating MTT to a particular situation unearths a previously unknown type theory that improves upon prior systems. Finally, we investigate the metatheory of MTT. We prove the consistency of MTT and establish canonicity through an extension of recent type-theoretic gluing techniques. These results hold irrespective of the choice of mode theory, and thus apply to a wide variety of modal situations.

cs.LO↗

arXiv2020

A Cubical Language for Bishop Sets

We present XTT, a version of Cartesian cubical type theory specialized for Bishop sets \`a la Coquand, in which every type enjoys a definitional version of the uniqueness of identity proofs. Using cubical notions, XTT reconstructs many of the ideas underlying Observational Type Theory, a version of intensional type theory that supports function extensionality. We prove the canonicity property of XTT (that every closed boolean is definitionally equal to a constant) using Artin gluing.

cs.LO↗

arXiv2019

Cubical Syntax for Reflection-Free Extensional Equality

We contribute XTT, a cubical reconstruction of Observational Type Theory which extends Martin-L\"of's intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental version of the unicity of identity types principle (UIP): any two elements of the same equality type are judgmentally equal. Moreover, we conjecture that the typing relation can be decided in a practical way. In this paper, we establish an algebraic canonicity theorem using a novel cubical extension (independently proposed by Awodey) of the logical families or categorical gluing argument inspired by Coquand and Shulman: every closed element of boolean type is derivably equal to either 'true' or 'false'.

cs.LO↗