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 Synthetic Tait Computability the Hard Way, ArXiv preprint. Exponentiable locales, revisited, ArXiv preprint. Recursion formula for knot invariants, Undergraduate Thesis. [Slides]. 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]. Mathieu, a puzzle whose configuration space is the sporadic group. ASKL, a 4K rhythm game supporting the Malody format. References 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 Related 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 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