https://trebor-huang.github.io/forest/trebor/ trebor /forest/trebor/ Trebor Person Indiana University, Bloomington PhD Student https://github.com/Trebor-Huang https://types.pl/@trebor xh40@iu.edu My name is 黄栩 (Huang, Xu), but I go by the name Trebor online. I’m interested in the syntax and semantics of dependent type theory, in particular homotopy type theory and cubical type theory. I am currently a PhD student at Indiana University Bloomington, supervised by Carlo Angiuli. I graduated from Tsinghua University. Writings Fat cell structures and generalized algebraic theories (with Carlo Angiuli), to appear in LICS 2026. Normal forms in Cubical Type Theory, ArXiv preprint. Synthetic Tait Computability the Hard Way, ArXiv preprint. Exponentiable locales, revisited, ArXiv preprint. Recursion formula for knot invariants, Undergraduate Thesis. [Slides]. I have a List of open problems that I’m interested in. I am also writing some expository material about type theory in Chinese. Mainly History of Type Theory and Models of Dependent Type Theory (unfinished). Random fun stuff This site itself is my forest, where I write about stuff I learned. Lie Sphere Geometry, an interesting geometry with symmetry where points, lines and circles are on the same footing. [GitHub repo]. ASKL, a 4K rhythm game supporting the Malody format. References Trebor 2026 3 26 https://trebor-huang.github.io/forest/cubical-normal-form/ cubical-normal-form /forest/cubical-normal-form/ Normal forms in Cubical Type Theory Reference 10.48550/arXiv.2603.24923 This note documents the specification of normal forms in cubical type theory. The definition is already present in the proof of normalization for cubical type theory, but we present it in a more traditional style explicitly for reference. Trebor 2025 6 21 https://trebor-huang.github.io/forest/exp-locale/ exp-locale /forest/exp-locale/ Exponentiable locales, revisited Reference 10.48550/arXiv.2507.15579 We give a moderately motivated exposition of exponentiable locales and the construction of exponentials in , without assuming prior knowledge of exponential topological spaces or continuous posets. Trebor 2025 5 25 https://trebor-huang.github.io/forest/history/ history /forest/history/ History of Type Theory Reference https://github.com/Trebor-Huang/history Trebor 2023 10 3 https://trebor-huang.github.io/forest/stc-hard-way/ stc-hard-way /forest/stc-hard-way/ Synthetic Tait Computability the Hard Way Reference 10.48550/arXiv.2310.02051 We walk through a few proofs of canonicity and normalization, each one with more aspects dissected and re-expressed in category theory, so that readers can compare the difference across proofs. During this process we isolate the different ideas that make up the proofs. Finally we arrive at synthetic Tait computability as proposed by J. Sterling. We also give a synthetic proof for parametricity of system F. Trebor https://trebor-huang.github.io/forest/models/ models /forest/models/ Models of Dependent Type Theory Reference https://github.com/Trebor-Huang/model Context Backlinks Trebor 2026 6 14 https://trebor-huang.github.io/forest/trebor-0003/ trebor-0003 /forest/trebor-0003/ List of open problems The following is a list of open (to my knowledge) problems that I’m interested in, with some thoughts on them. They are mostly in type theory, constructive logic and computability theory. I think about them from time to time, and if any problem is already solved in the literature, or has a new claimed solution, please contact me! Trebor2026614https://trebor-huang.github.io/forest/open-0001/open-0001/forest/open-0001/Strongly total functions on Church numeralsQuestion What are all the total functions on natural numbers such that there exists an untyped -calculus term such that (where is the -th Church numeral) is strongly normalizing for each , and evaluates to ? This is raised on MathOverflow. In some sense, this is asking for a kind of limit on type systems that guarantee strong normalization. This class of total functions includes for example all the functions of type in system , where is encoded as . These are the functions provably total in second order Peano arithmetic. Similarly system gives all the functions provably total in higher order logic. These are all we get out of the -cube. However, note that it’s not necessarily true that system can only provide us with these functions. It is possible that a term has the property that each is typable (hence strongly normalizing) in system , but the types assigned to or depends on . I’m also interested to see if there are these wild total functions. Perhaps one direction of attack is to consider higher order logic equipped with stronger and stronger large cardinal axioms. We then somehow translate them to strongly normalizing type systems (with more and more universes). Trebor2026614https://trebor-huang.github.io/forest/open-0002/open-0002/forest/open-0002/Dimensionlity of the rational number localeQuestion Classify the locales up to homeomorphism. This is raised on MathOverflow. It is well-known that is not homeomorphic to as locales, which is in stark contrast with topological spaces where are all homeomorphic. The proof is a delicate analysis on rectangles on the rational plane. Perhaps similar arguments can be carried out on higher dimensional spaces, but I haven’t given too much thought to it. Trebor2026615https://trebor-huang.github.io/forest/open-0003/open-0003/forest/open-0003/-categories in vanilla HoTTQuestion Formalize, or prove the impossibility for a definition of -categories in the style of synthetic homotopy theory in HoTT. Defining -categories following the usual, classical definition in HoTT is generally considered unproblematic, and just a matter of effort. There has been various projects working towards this. A definition of -categories synthetically should have the correct interpretation in the canonical model of HoTT in -groupoids (or more concretely, simplicial sets), up to equivalence. We should also be able to extract the -types, and define composition and the -associator for each concrete number . More generally, the axioms developed by Cisinski and others should be provable. There has been various approaches to defining -categories in extensions of HoTT. They generally correspond to the two styles of defining categories: Defining -sets as families of sets over the collection of objects. Generalizing to -categories requires arbitrarily long chains of dependencies. This is the idea of the work of displayed type theory. Defining the total collection of arrows, with two projections and . Generalizing to -categories requires the definition of (semi-)simplicial spaces, which require an infinite tower of coherences between these projection maps. This leads to the work of two-level type theory. Trebor2026615https://trebor-huang.github.io/forest/open-0004/open-0004/forest/open-0004/The theory of presentable and accessible categories in type theoryQuestion Is it possible to develop the theory of presentable and accessible categories in a constructive (or even predicative) type theory? Barring this, how can we develop this theory in a type-theory idiomatic way? It is known that the classical theory of presentable and accessible categories heavily relies on the axiom of choice. There are several questions on MathOverflow on this topic: Locally presentable and accessible categories without the axiom of choice? Accessibility of small categories without choice Obstructions to the theory of locally presentable categories without choice We can avoid a lot of choice unique up to unique isomorphism in a univalent type theory. It may also be possible to redefine notions of size using an inductive-recursively defined Tarski universe. I suspect this to be easier to use than trying to develop a theory of cardinals. Trebor2026615https://trebor-huang.github.io/forest/open-0005/open-0005/forest/open-0005/Recursively realizable one-variable formula in propositional logicQuestion What are all the propositional formulas involving one propositional variable that are realizable with recursive functions? In other words, what propositions are true in the computable world? Constructive logic is realizable with computable functions, and so we can restrict our attention to the free Heyting algebra generated by one element, also known as the Rieger–Nishimura lattice, and asking if any proposition other than is realizable. According to this survey, although there are propositions unprovable in constructive logic but realizable computably, the existence of such one-variable propositions is still open. Trebor2026615https://trebor-huang.github.io/forest/open-0006/open-0006/forest/open-0006/The orthogonal and unitary groups in synthetic homotopy theoryQuestion Define the orthogonal groups and the unitary groups in synthetic homotopy theory, together with their actions on the spheres of corresponding dimensions. Their deloopings as well as the direct limits , etc. are also of interest. These definitions are necessary for the study of Bott periodicity, as well as the -homomorphism in the homotopy groups of spheres. There has been work on defining the tangent unit sphere bundle of spheres. It is plausible that we can build on this construction to define the orthogonal group. It is however unclear how we would define the group operation. See also these slides. Trebor2026615https://trebor-huang.github.io/forest/open-0007/open-0007/forest/open-0007/Meta-theory of complex inductive definitionsQuestion Specify a schema of inductive definitions that can handle arbitrary interleaving, nesting and dependency of induction and recursion. Prove or disprove the consistency, canonicity and normalization of these inductive definitions, as well as their compatibility with excluded middle, the axiom of choice and univalence. Agda is extremely permissive on the validity of inductive definitions, far surpassing any literature. The declaration and definitions can be arbitrarily interleaved, and later definitions can refer to previous declarations freely (subject to termination checking). The strict-positivity checking is an “open” condition, meaning that each new feature of the type theory adds to the possible inductive definitions. So a “closed” approach such as reducing inductive types to -types seems inappropriate. Some of these inductive types have been found to be inconsistent, or incompatible with excluded middle or univalence. However, fixing these inconsistencies reactively is not an ideal solution. Given that a large portion of these definitions seems intuitive, it should be possible to establish at least their consistency relative to set theory with well-known large cardinal axioms. See also this StackExchange question. Trebor2026615https://trebor-huang.github.io/forest/open-0008/open-0008/forest/open-0008/Higher effective toposQuestion Define an elementary -topos version of the effective topos. In particular, the -truncated objects should recover exactly the effective topos. There has been a line of work defining and working with the effective 2-topos. On the other hand, trying to directly define assemblies in cubical sets gives the cubical assemblies, which unfortunately does not satisfy some principles that we expect of the effective topos, such as the Church thesis. There are ways to fix this problem, but it is ad hoc and doesn’t seem to be satisfactory as an answer to the question. Trebor2026615https://trebor-huang.github.io/forest/open-0009/open-0009/forest/open-0009/Computational interpretation of propositional resizingQuestion Give a computational interpretation of propositional resizing in homotopy type theory. A naïve approach does not work here, since the computation needs to somehow get stuck on the proof of propositionality. Otherwise, we can assume (falsely) that any type is a proposition, which allows us to replicate Russell’s paradox and produce non-terminating reduction in a false context. This is related to Question . Related Trebor 2026 6 14 https://trebor-huang.github.io/forest/trebor-0003/ trebor-0003 /forest/trebor-0003/ List of open problems The following is a list of open (to my knowledge) problems that I’m interested in, with some thoughts on them. They are mostly in type theory, constructive logic and computability theory. I think about them from time to time, and if any problem is already solved in the literature, or has a new claimed solution, please contact me! Trebor2026614https://trebor-huang.github.io/forest/open-0001/open-0001/forest/open-0001/Strongly total functions on Church numeralsQuestion What are all the total functions on natural numbers such that there exists an untyped -calculus term such that (where is the -th Church numeral) is strongly normalizing for each , and evaluates to ? This is raised on MathOverflow. In some sense, this is asking for a kind of limit on type systems that guarantee strong normalization. This class of total functions includes for example all the functions of type in system , where is encoded as . These are the functions provably total in second order Peano arithmetic. Similarly system gives all the functions provably total in higher order logic. These are all we get out of the -cube. However, note that it’s not necessarily true that system can only provide us with these functions. It is possible that a term has the property that each is typable (hence strongly normalizing) in system , but the types assigned to or depends on . I’m also interested to see if there are these wild total functions. Perhaps one direction of attack is to consider higher order logic equipped with stronger and stronger large cardinal axioms. We then somehow translate them to strongly normalizing type systems (with more and more universes). Trebor2026614https://trebor-huang.github.io/forest/open-0002/open-0002/forest/open-0002/Dimensionlity of the rational number localeQuestion Classify the locales up to homeomorphism. This is raised on MathOverflow. It is well-known that is not homeomorphic to as locales, which is in stark contrast with topological spaces where are all homeomorphic. The proof is a delicate analysis on rectangles on the rational plane. Perhaps similar arguments can be carried out on higher dimensional spaces, but I haven’t given too much thought to it. Trebor2026615https://trebor-huang.github.io/forest/open-0003/open-0003/forest/open-0003/-categories in vanilla HoTTQuestion Formalize, or prove the impossibility for a definition of -categories in the style of synthetic homotopy theory in HoTT. Defining -categories following the usual, classical definition in HoTT is generally considered unproblematic, and just a matter of effort. There has been various projects working towards this. A definition of -categories synthetically should have the correct interpretation in the canonical model of HoTT in -groupoids (or more concretely, simplicial sets), up to equivalence. We should also be able to extract the -types, and define composition and the -associator for each concrete number . More generally, the axioms developed by Cisinski and others should be provable. There has been various approaches to defining -categories in extensions of HoTT. They generally correspond to the two styles of defining categories: Defining -sets as families of sets over the collection of objects. Generalizing to -categories requires arbitrarily long chains of dependencies. This is the idea of the work of displayed type theory. Defining the total collection of arrows, with two projections and . Generalizing to -categories requires the definition of (semi-)simplicial spaces, which require an infinite tower of coherences between these projection maps. This leads to the work of two-level type theory. Trebor2026615https://trebor-huang.github.io/forest/open-0004/open-0004/forest/open-0004/The theory of presentable and accessible categories in type theoryQuestion Is it possible to develop the theory of presentable and accessible categories in a constructive (or even predicative) type theory? Barring this, how can we develop this theory in a type-theory idiomatic way? It is known that the classical theory of presentable and accessible categories heavily relies on the axiom of choice. There are several questions on MathOverflow on this topic: Locally presentable and accessible categories without the axiom of choice? Accessibility of small categories without choice Obstructions to the theory of locally presentable categories without choice We can avoid a lot of choice unique up to unique isomorphism in a univalent type theory. It may also be possible to redefine notions of size using an inductive-recursively defined Tarski universe. I suspect this to be easier to use than trying to develop a theory of cardinals. Trebor2026615https://trebor-huang.github.io/forest/open-0005/open-0005/forest/open-0005/Recursively realizable one-variable formula in propositional logicQuestion What are all the propositional formulas involving one propositional variable that are realizable with recursive functions? In other words, what propositions are true in the computable world? Constructive logic is realizable with computable functions, and so we can restrict our attention to the free Heyting algebra generated by one element, also known as the Rieger–Nishimura lattice, and asking if any proposition other than is realizable. According to this survey, although there are propositions unprovable in constructive logic but realizable computably, the existence of such one-variable propositions is still open. Trebor2026615https://trebor-huang.github.io/forest/open-0006/open-0006/forest/open-0006/The orthogonal and unitary groups in synthetic homotopy theoryQuestion Define the orthogonal groups and the unitary groups in synthetic homotopy theory, together with their actions on the spheres of corresponding dimensions. Their deloopings as well as the direct limits , etc. are also of interest. These definitions are necessary for the study of Bott periodicity, as well as the -homomorphism in the homotopy groups of spheres. There has been work on defining the tangent unit sphere bundle of spheres. It is plausible that we can build on this construction to define the orthogonal group. It is however unclear how we would define the group operation. See also these slides. Trebor2026615https://trebor-huang.github.io/forest/open-0007/open-0007/forest/open-0007/Meta-theory of complex inductive definitionsQuestion Specify a schema of inductive definitions that can handle arbitrary interleaving, nesting and dependency of induction and recursion. Prove or disprove the consistency, canonicity and normalization of these inductive definitions, as well as their compatibility with excluded middle, the axiom of choice and univalence. Agda is extremely permissive on the validity of inductive definitions, far surpassing any literature. The declaration and definitions can be arbitrarily interleaved, and later definitions can refer to previous declarations freely (subject to termination checking). The strict-positivity checking is an “open” condition, meaning that each new feature of the type theory adds to the possible inductive definitions. So a “closed” approach such as reducing inductive types to -types seems inappropriate. Some of these inductive types have been found to be inconsistent, or incompatible with excluded middle or univalence. However, fixing these inconsistencies reactively is not an ideal solution. Given that a large portion of these definitions seems intuitive, it should be possible to establish at least their consistency relative to set theory with well-known large cardinal axioms. See also this StackExchange question. Trebor2026615https://trebor-huang.github.io/forest/open-0008/open-0008/forest/open-0008/Higher effective toposQuestion Define an elementary -topos version of the effective topos. In particular, the -truncated objects should recover exactly the effective topos. There has been a line of work defining and working with the effective 2-topos. On the other hand, trying to directly define assemblies in cubical sets gives the cubical assemblies, which unfortunately does not satisfy some principles that we expect of the effective topos, such as the Church thesis. There are ways to fix this problem, but it is ad hoc and doesn’t seem to be satisfactory as an answer to the question. Trebor2026615https://trebor-huang.github.io/forest/open-0009/open-0009/forest/open-0009/Computational interpretation of propositional resizingQuestion Give a computational interpretation of propositional resizing in homotopy type theory. A naïve approach does not work here, since the computation needs to somehow get stuck on the proof of propositionality. Otherwise, we can assume (falsely) that any type is a proposition, which allows us to replicate Russell’s paradox and produce non-terminating reduction in a false context. This is related to Question . Trebor 2024 2 16 https://trebor-huang.github.io/forest/trebor-0001/ trebor-0001 /forest/trebor-0001/ Trebor’s forest This is my forest. Forests are a way of organizing notes. Some of the contents here: Generating aperiodic tilings with substitution systems Notes on homological algebra Effective homology Algebraic geometry (Under namespace algm, not organized) Beck monadicity theorem On Ghost and phantom maps Spectral sequence of a tower of fibrations On a computer, you can press Ctrl–K to search for trees. Trebor 2024 3 4 https://trebor-huang.github.io/forest/trebor-0002/ trebor-0002 /forest/trebor-0002/ Style of writing I try to adhere to several principles when composing trees. If a definition or a lemma would present itself during a proof attempt, then do not introduce it until that part of the proof. The nature of forests makes this kind of organization more viable, as each definition is a tree by itself. If a part of a proof has an easy constructive phrasing, use it and avoid classical reasoning. But if it requires some complication, then the classical way is used. Avoid abstraction leaks. Even if A is defined as B, do not use them interchangeably. For instance, an ideal is defined as a subset of a ring satisfying certain conditions. But we will insist on writing ideals as numbers, for example saying instead of . This is not too heretic, since algebraic number theory already uses this kind of notation. Similarly, although prime ideals correspond to points in , we do not identify them. Apply unbiased terminology and negative thinking. https://trebor-huang.github.io/forest/angiuli/ angiuli /forest/angiuli/ Carlo Angiuli Person Indiana University, Bloomington Assistant Professor https://carloangiuli.com/ Contributions Trebor 2026 3 26 https://trebor-huang.github.io/forest/cubical-normal-form/ cubical-normal-form /forest/cubical-normal-form/ Normal forms in Cubical Type Theory Reference 10.48550/arXiv.2603.24923 This note documents the specification of normal forms in cubical type theory. The definition is already present in the proof of normalization for cubical type theory, but we present it in a more traditional style explicitly for reference. Trebor 2025 6 21 https://trebor-huang.github.io/forest/exp-locale/ exp-locale /forest/exp-locale/ Exponentiable locales, revisited Reference 10.48550/arXiv.2507.15579 We give a moderately motivated exposition of exponentiable locales and the construction of exponentials in , without assuming prior knowledge of exponential topological spaces or continuous posets. Trebor 2025 5 25 https://trebor-huang.github.io/forest/history/ history /forest/history/ History of Type Theory Reference https://github.com/Trebor-Huang/history Trebor 2023 10 3 https://trebor-huang.github.io/forest/stc-hard-way/ stc-hard-way /forest/stc-hard-way/ Synthetic Tait Computability the Hard Way Reference 10.48550/arXiv.2310.02051 We walk through a few proofs of canonicity and normalization, each one with more aspects dissected and re-expressed in category theory, so that readers can compare the difference across proofs. During this process we isolate the different ideas that make up the proofs. Finally we arrive at synthetic Tait computability as proposed by J. Sterling. We also give a synthetic proof for parametricity of system F. Trebor https://trebor-huang.github.io/forest/models/ models /forest/models/ Models of Dependent Type Theory Reference https://github.com/Trebor-Huang/model