Internal lenses as functors and cofunctors

Bryce Clarke

doi:10.4204/EPTCS.323.13

Internal lenses as functors and cofunctors

Bryce Clarke^*

^*Corresponding author for this work

Research output: Contribution to journal › Conference paper

7 Citations (Scopus)

25 Downloads (Pure)

Abstract

Lenses may be characterised as objects in the category of algebras over a monad, however they are often understood instead as morphisms, which propagate updates between systems. Working internally to a category with pullbacks, we define lenses as simultaneously functors and cofunctors between categories. We show that lenses may be canonically represented as a particular commuting triangle of functors, and unify the classical state-based lenses with both c-lenses and d-lenses in this framework. This new treatment of lenses leads to considerable simplifications that are important in applications, including a clear interpretation of lens composition.

Original language	English
Pages (from-to)	183-195
Number of pages	13
Journal	Electronic Proceedings in Theoretical Computer Science, EPTCS
Volume	323
DOIs	https://doi.org/10.4204/EPTCS.323.13
Publication status	Published - 15 Sept 2020
Event	2019 Applied Category Theory 2019, ACT 2019 - Oxford, United Kingdom Duration: 15 Jul 2019 → 19 Jul 2019

Bibliographical note

Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

Journal issue edited by: John Baez and Bob Coecke

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10.4204/EPTCS.323.13Licence: CC BY

Publisher version (open access)Final published version, 161 KBLicence: CC BY

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N2 - Lenses may be characterised as objects in the category of algebras over a monad, however they are often understood instead as morphisms, which propagate updates between systems. Working internally to a category with pullbacks, we define lenses as simultaneously functors and cofunctors between categories. We show that lenses may be canonically represented as a particular commuting triangle of functors, and unify the classical state-based lenses with both c-lenses and d-lenses in this framework. This new treatment of lenses leads to considerable simplifications that are important in applications, including a clear interpretation of lens composition.

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JF - Electronic Proceedings in Theoretical Computer Science, EPTCS

T2 - 2019 Applied Category Theory 2019, ACT 2019

Y2 - 15 July 2019 through 19 July 2019

ER -

Internal lenses as functors and cofunctors

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