A Fibrational Theory of First Order Differential Structures

Reference. A Fibrational Theory of First Order Differential Structures

We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation, including cartesian differential categories, generalised cartesian differential categories, tangent categories, as well as the versions of these categories axiomatising reverse derivatives. We explain uniformly and concisely the requirements expressed by these structures, using sections of suitable fibrations as unifying concept. Our perspective sheds light on their similarities and differences, as well as simplifying certain constructions from the literature.

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Cite as @capucci-2024-a (helia, typst) · \cite{capucci-2024-a} (LaTeX)

BibTeX

bibtex · 10 lines

@misc{capucci-2024-a,
  doi = {10.48550/ARXIV.2409.05763},
  url = {https://arxiv.org/abs/2409.05763},
  author = {Capucci, Matteo and Cruttwell, Geoffrey S. H. and Ghani, Neil and Zanasi, Fabio},
  keywords = {Category Theory (math.CT), Logic in Computer Science (cs.LO), FOS: Mathematics, FOS: Mathematics, FOS: Computer and information sciences, FOS: Computer and information sciences},
  title = {A Fibrational Theory of First Order Differential Structures},
  publisher = {arXiv},
  year = {2024},
  copyright = {Creative Commons Attribution 4.0 International}
}

hayagriva YAML (typst)

yaml · 13 lines

capucci-2024-a:
  type: misc
  title: A Fibrational Theory of First Order Differential Structures
  author:
  - Capucci, Matteo
  - Cruttwell, Geoffrey S. H.
  - Ghani, Neil
  - Zanasi, Fabio
  date: 2024
  publisher: arXiv
  url: https://arxiv.org/abs/2409.05763
  serial-number:
    doi: 10.48550/ARXIV.2409.05763

Download: capucci-2024-a.bib · capucci-2024-a.yml

Cites 32 works (1 here)

With notes (1)

Categorical Logic and Type Theory jacobs-19991999

This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.

External (31)

Reverse Tangent Categories (2024)
Differential Bundles in Commutative Algebra and Algebraic Geometry (2023)
Monoidal reverse differential categories (2022)
Functorial String Diagrams for Reverse-Mode Automatic Differentiation (2021)
Categorical Foundations of Gradient-Based Learning (2021)
Poly: An abundant categorical setting for mode-dependent dynamics (2020)
Supplying bells and whistles in symmetric monoidal categories (2020)
Reverse derivative categories (2019)
Generalized Lens Categories via functors $\mathcalC^ \rm op\to \mathsfCat $ (2019)
lens: Lenses, folds and traversals (Haskell library) (2019)
Fibered Categories a la Jean Benabou (2018)
Coherence for lenses and open games (2017)
Connections in tangent categories (2016)
Differential bundles and fibrations for tangent categories (2016)
An introduction to differential linear logic: proof-nets, models and antiderivatives (2016)
Differential Structure, Tangent Structure, and SDG (2014)
Polynomial functors and polynomial monads (2013)
Cartesian differential categories revisited (2012)
Who invented the reverse mode of differentiation? (2012)
Cartesian Differential Categories (2009)
Differential categories (2006)
Containers: Constructing strictly positive types (2005)
The differential lambda-calculus (2003)
Proper factorization systems in 2-categories (2003)
Some properties of Fib as a fibred 2-category (1999)
Practical Foundations of Mathematics (1999)
Elementary observations on 2-categorical limits (1989)
Abstract tangent functors (1984)
Update semantics of relational views (1981)
Coalgebras and cartesian categories (1976)
Fibrations and Yoneda's lemma in a 2-category (1974)