Cartesian Difference Categories
Cartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential $\lambda...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Logical Methods in Computer Science e.V.
2021-09-01
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| Series: | Logical Methods in Computer Science |
| Subjects: | |
| Online Access: | https://lmcs.episciences.org/6924/pdf |
| _version_ | 1797268503931125760 |
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| author | Mario Alvarez-Picallo Jean-Simon Pacaud Lemay |
| author_facet | Mario Alvarez-Picallo Jean-Simon Pacaud Lemay |
| author_sort | Mario Alvarez-Picallo |
| collection | DOAJ |
| description | Cartesian differential categories are categories equipped with a differential
combinator which axiomatizes the directional derivative. Important models of
Cartesian differential categories include classical differential calculus of
smooth functions and categorical models of the differential $\lambda$-calculus.
However, Cartesian differential categories cannot account for other interesting
notions of differentiation of a more discrete nature such as the calculus of
finite differences. On the other hand, change action models have been shown to
capture these examples as well as more "exotic" examples of differentiation.
But change action models are very general and do not share the nice properties
of Cartesian differential categories. In this paper, we introduce Cartesian
difference categories as a bridge between Cartesian differential categories and
change action models. We show that every Cartesian differential category is a
Cartesian difference category, and how certain well-behaved change action
models are Cartesian difference categories. In particular, Cartesian difference
categories model both the differential calculus of smooth functions and the
calculus of finite differences. Furthermore, every Cartesian difference
category comes equipped with a tangent bundle monad whose Kleisli category is
again a Cartesian difference category. |
| first_indexed | 2024-04-25T01:33:31Z |
| format | Article |
| id | doaj.art-89ee3076998747c09a932022fb1eda30 |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-25T01:33:31Z |
| publishDate | 2021-09-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-89ee3076998747c09a932022fb1eda302024-03-08T10:35:15ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742021-09-01Volume 17, Issue 310.46298/lmcs-17(3:23)20216924Cartesian Difference CategoriesMario Alvarez-PicalloJean-Simon Pacaud LemayCartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential $\lambda$-calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation of a more discrete nature such as the calculus of finite differences. On the other hand, change action models have been shown to capture these examples as well as more "exotic" examples of differentiation. But change action models are very general and do not share the nice properties of Cartesian differential categories. In this paper, we introduce Cartesian difference categories as a bridge between Cartesian differential categories and change action models. We show that every Cartesian differential category is a Cartesian difference category, and how certain well-behaved change action models are Cartesian difference categories. In particular, Cartesian difference categories model both the differential calculus of smooth functions and the calculus of finite differences. Furthermore, every Cartesian difference category comes equipped with a tangent bundle monad whose Kleisli category is again a Cartesian difference category.https://lmcs.episciences.org/6924/pdfmathematics - category theorycomputer science - logic in computer science |
| spellingShingle | Mario Alvarez-Picallo Jean-Simon Pacaud Lemay Cartesian Difference Categories Logical Methods in Computer Science mathematics - category theory computer science - logic in computer science |
| title | Cartesian Difference Categories |
| title_full | Cartesian Difference Categories |
| title_fullStr | Cartesian Difference Categories |
| title_full_unstemmed | Cartesian Difference Categories |
| title_short | Cartesian Difference Categories |
| title_sort | cartesian difference categories |
| topic | mathematics - category theory computer science - logic in computer science |
| url | https://lmcs.episciences.org/6924/pdf |
| work_keys_str_mv | AT marioalvarezpicallo cartesiandifferencecategories AT jeansimonpacaudlemay cartesiandifferencecategories |