Internal lenses as functors and cofunctors

@inproceedings{Clarke2020InternalLA,
  title={Internal lenses as functors and cofunctors},
  author={Bryce Clarke},
  year={2020}
}
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… 
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