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.
|Number of pages||13|
|Journal||Electronic Proceedings in Theoretical Computer Science, EPTCS|
|Publication status||Published - 15 Sep 2020|
|Event||2019 Applied Category Theory 2019, ACT 2019 - Oxford, United Kingdom|
Duration: 15 Jul 2019 → 19 Jul 2019
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Journal issue edited by: John Baez and Bob Coecke