arXiv subjects

Ulrik Buchholtz

Publications and source records attributed to Ulrik Buchholtz.

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.

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arXiv2026

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.

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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.

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arXiv2025

Central H-spaces and banded types

We introduce and study central types, which are generalizations of Eilenberg-Mac Lane spaces. A type is central when it is equivalent to the component of the identity among its own self-equivalences. From centrality alone we construct an infinite delooping in terms of a tensor product of banded types, which are the appropriate notion of torsor for a central type. Our constructions are carried out in homotopy type theory, and therefore hold in any $\infty$-topos. Even when interpreted into the $\infty$-topos of spaces, our approach to constructing these deloopings is new. Along the way, we further develop the theory of H-spaces in homotopy type theory, including their relation to evaluation fibrations and Whitehead products. These considerations let us, for example, rule out the existence of H-space structures on the $2n$-sphere for $n > 0$. We also give a novel description of the moduli space of H-space structures on an H-space. Using this description, we generalize a formula of Arkowitz-Curjel and Copeland for counting the number of path components of this moduli space. As an application, we deduce that the moduli space of H-space structures on the $3$-sphere is $Ω^6 \mathbb{S}^3$.

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arXiv2024

Epimorphisms and Acyclic Types in Univalent Foundations

We characterize the epimorphisms in homotopy type theory (HoTT) as the fiberwise acyclic maps and develop a type-theoretic treatment of acyclic maps and types in the context of synthetic homotopy theory as developed in univalent foundations. We present examples and applications in group theory, such as the acyclicity of the Higman group, through the identification of groups with 0-connected, pointed 1-types. Many of our results are formalized as part of the agda-unimath library.

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arXiv2024

Primitive Recursive Dependent Type Theory

We show that restricting the elimination principle of the natural numbers type in Martin-Löf Type Theory (MLTT) to a universe of types not containing $Π$-types ensures that all definable functions are primitive recursive. This extends the concept of primitive recursiveness to general types. We discuss extensions to univalent type theories and other notions of computability. We are inspired by earlier work by Martin Hofmann, work on Joyal's arithmetic universes, and Hugo Herbelin and Ludovic Patey's sketched Calculus of Primitive Recursive Constructions. We define a theory Tpr that is a subtheory of MLTT with two universes, such that all inductive types are finitary and the lowest universe is restricted to not contain $Π$-types. We prove soundness such that all functions $\mathbb{N}\to\mathbb{N}$ are primitive recursive. The proof requires that Tpr satisfies canonicity, which we easily prove using synthetic Tait computability.

math.LO↗

arXiv2024

On symmetries of spheres in univalent foundations

Working in univalent foundations, we investigate the symmetries of spheres, i.e., the types of the form $\mathbb{S}^n = \mathbb{S}^n$. The case of the circle has a slick answer: the symmetries of the circle form two copies of the circle. For higher-dimensional spheres, the type of symmetries has again two connected components, namely the components of the maps of degree plus or minus one. Each of the two components has $\mathbb{Z}/2\mathbb{Z}$ as fundamental group. For the latter result, we develop an EHP long exact sequence.

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arXiv2022

Synthetic fibered $(\infty,1)$-category theory

We study cocartesian fibrations in the setting of the synthetic $(\infty,1)$-category theory developed in the simplicial type theory introduced by Riehl and Shulman. Our development culminates in a Yoneda Lemma for cocartesian fibrations.

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arXiv2021

The long exact sequence of homotopy $n$-groups

Working in homotopy type theory, we introduce the notion of $n$-exactness for a short sequence $F\to E\to B$ of pointed types, and show that any fiber sequence $F\hookrightarrow E \twoheadrightarrow B$ of arbitrary types induces a short sequence $\|F\|_{n-1} \to \|E\|_{n-1} \to \|B\|_{n-1}$ that is $n$-exact at $\|E\|_{n-1}$. We explain how the indexing makes sense when interpreted in terms of $n$-groups, and we compare our definition to the existing definitions of an exact sequence of $n$-groups for $n=1,2$. As the main application, we obtain the long $n$-exact sequence of homotopy $n$-groups of a fiber sequence.

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arXiv2020

Construction of the Circle in UniMath

We show that the type $\mathrm{T}\mathbb{Z}$ of $\mathbb{Z}$-torsors has the dependent universal property of the circle, which characterizes it up to a unique homotopy equivalence. The construction uses Voevodsky's Univalence Axiom and propositional truncation, yielding a stand-alone construction of the circle not using higher inductive types.

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arXiv2020

Cellular Cohomology in Homotopy Type Theory

We present a development of cellular cohomology in homotopy type theory. Cohomology associates to each space a sequence of abelian groups capturing part of its structure, and has the advantage over homotopy groups in that these abelian groups of many common spaces are easier to compute. Cellular cohomology is a special kind of cohomology designed for cell complexes: these are built in stages by attaching spheres of progressively higher dimension, and cellular cohomology defines the groups out of the combinatorial description of how spheres are attached. Our main result is that for finite cell complexes, a wide class of cohomology theories (including the ones defined through Eilenberg-MacLane spaces) can be calculated via cellular cohomology. This result was formalized in the Agda proof assistant.

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arXiv2018

Higher Structures in Homotopy Type Theory

The intended model of the homotopy type theories used in Univalent Foundations is the infinity-category of homotopy types, also known as infinity-groupoids. The problem of higher structures is that of constructing the homotopy types needed for mathematics, especially those that aren't sets. The current repertoire of constructions, including the usual type formers and higher inductive types, suffice for many but not all of these. We discuss the problematic cases, typically those involving an infinite hierarchy of coherence data such as semi-simplicial types, as well as the problem of developing the meta-theory of homotopy type theories in Univalent Foundations. We also discuss some proposed solutions.

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arXiv2018

Higher Groups in Homotopy Type Theory

We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure inherent in the identity types of Martin-Löf type theory. We investigate ordinary groups from this viewpoint, as well as higher dimensional groups and groups that can be delooped more than once. A major result is the stabilization theorem, which states that if an $n$-type can be delooped $n+2$ times, then it is an infinite loop type. Most of the results have been formalized in the Lean proof assistant.

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arXiv2017

Syntactic Forcing Models for Coherent Logic

We present three syntactic forcing models for coherent logic. These are based on sites whose underlying category only depends on the signature of the coherent theory, and they do not presuppose that the logic has equality. As an application we give a coherent theory T and a sentence ψ which is T-redundant (for any geometric implication ϕ, possibly with equality, if T + ψ proves ϕ, then T proves ϕ), yet false in the generic model of T. This answers in the negative a question by Wraith.

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arXiv2017

Homotopy Type Theory in Lean

We discuss the homotopy type theory library in the Lean proof assistant. The library is especially geared toward synthetic homotopy theory. Of particular interest is the use of just a few primitive notions of higher inductive types, namely quotients and truncations, and the use of cubical methods.

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arXiv2017

Varieties of Cubical Sets

We define a variety of notions of cubical sets, based on sites organized using substructural algebraic theories presenting PRO(P)s or Lawvere theories. We prove that all our sites are test categories in the sense of Grothendieck, meaning that the corresponding presheaf categories of cubical sets model classical homotopy theory. We delineate exactly which ones are even strict test categories, meaning that products of cubical sets correspond to products of homotopy types.

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arXiv2017

The real projective spaces in homotopy type theory

Homotopy type theory is a version of Martin-Löf type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key players in homotopy theory, as certain higher inductive types in homotopy type theory. The classical definition of RP(n), as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define RP(n) by induction on n simultaneously with its tautological bundle of 2-element sets. As the base case, we take RP(-1) to be the empty type. In the inductive step, we take RP(n+1) to be the mapping cone of the projection map of the tautological bundle of RP(n), and we use its universal property and the univalence axiom to define the tautological bundle on RP(n+1). By showing that the total space of the tautological bundle of RP(n) is the n-sphere, we retrieve the classical description of RP(n+1) as RP(n) with an (n+1)-cell attached to it. The infinite dimensional real projective space, defined as the sequential colimit of the RP(n) with the canonical inclusion maps, is equivalent to the Eilenberg-MacLane space K(Z/2Z,1), which here arises as the subtype of the universe consisting of 2-element types. Indeed, the infinite dimensional projective space classifies the 0-sphere bundles, which one can think of as synthetic line bundles. These constructions in homotopy type theory further illustrate the utility of homotopy type theory, including the interplay of type theoretic and homotopy theoretic ideas.

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arXiv2016

The Cayley-Dickson Construction in Homotopy Type Theory

We define in the setting of homotopy type theory an H-space structure on $\mathbb S^3$. Hence we obtain a description of the quaternionic Hopf fibration $\mathbb S^3\hookrightarrow\mathbb S^7\twoheadrightarrow\mathbb S^4$, using only homotopy invariant tools.

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