Freely Generated Categorical Structures and Automatic Differentiation, PhD Thesis (Introduction and Conclusion)
Abstract
This version contains the introduction and conclusion of my PhD thesis, "Freely Generated Categorical Structures and Automatic Differentiation", together with its English and Dutch summaries. The full thesis consists of an introductory chapter, six joint research papers, and a concluding chapter, developed during my PhD studies at Utrecht University under the supervision of Gabriele Keller and Matthijs Vákár. The research papers are available separately and are not reproduced here. The introduction presents the scope of the thesis, explains the contributions of the six papers and their connections, and introduces the categorical foundations of our approach. The guiding idea is that programming languages, viewed as freely generated categorical structures, provide a principled setting for constructing structure-preserving program transformations and proving their correctness. Automatic differentiation supplies the central application: we study forward- and reverse-mode differentiation for expressive typed languages, including higher-order functions, recursive types, iteration and partiality. The semantic requirements of these transformations also motivate independent mathematical results on free distributive and extensive categories, cartesian closedness, and Grothendieck constructions. The conclusion brings these contributions together, discusses their limitations, and outlines further directions. Throughout, the thesis develops a dialogue between theory and practice: categorical semantics guides the construction of reliable and practically useful program transformations, while the demands of computation lead to new categorical structures and results.
Related
- Univalent Non-Associative Categorical Structures
- A Non-Dichotomy for CSPs of Stable ω-Categorical Structures
- Weight structures and t-structures on the triangulated category generated by a ℙ-object
- The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups
- Removing Boundary Smoothness from the Hot-Spots Conclusion in the Simple-Eigenvalue Case
BibTeX
@misc{nunes2026freely,
title = {{Freely Generated Categorical Structures and Automatic Differentiation, PhD Thesis (Introduction and Conclusion)}},
author = {Fernando Lucatelli Nunes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40046},
eprint = {2609.40046},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40046},
}
RIS
TY - UNPB
TI - Freely Generated Categorical Structures and Automatic Differentiation, PhD Thesis (Introduction and Conclusion)
AU - Fernando Lucatelli Nunes
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40046
UR - https://arxiv.org/abs/2609.40046
AB - This version contains the introduction and conclusion of my PhD thesis, "Freely Generated Categorical Structures and Automatic Differentiation", together with its English and Dutch summaries. The full thesis consists of an introductory chapter, six joint research papers, and a concluding chapter, developed during my PhD studies at Utrecht University under the supervision of Gabriele Keller and Matthijs Vákár. The research papers are available separately and are not reproduced here. The introduction presents the scope of the thesis, explains the contributions of the six papers and their connections, and introduces the categorical foundations of our approach. The guiding idea is that programming languages, viewed as freely generated categorical structures, provide a principled setting for constructing structure-preserving program transformations and proving their correctness. Automatic differentiation supplies the central application: we study forward- and reverse-mode differentiation for expressive typed languages, including higher-order functions, recursive types, iteration and partiality. The semantic requirements of these transformations also motivate independent mathematical results on free distributive and extensive categories, cartesian closedness, and Grothendieck constructions. The conclusion brings these contributions together, discusses their limitations, and outlines further directions. Throughout, the thesis develops a dialogue between theory and practice: categorical semantics guides the construction of reliable and practically useful program transformations, while the demands of computation lead to new categorical structures and results.
ER -