In [[Category Theory]], a lens is a concept that provides a way to focus on a specific part of an object within a category. It is used to modify or access specific components of an object, while preserving the structure of the object itself. A lens consists of two main components: a getter function and a setter function. The getter function retrieves the focused component from the object, while the setter function updates the focused component and returns a modified object. Formally, given a category C and an object A in C, a lens from A to B (written as Lens A B) represents a pair of functions: - get: A → B, which extracts the focused component from A - set: (A × B) → A, which takes an updated value for the focused component along with the original object and returns a new modified object The lens composition operator (∘) allows for chaining multiple lenses together to form complex lenses. This composition is analogous to function composition in other domains. Lenses provide a powerful way to work with immutable data structures in functional programming languages. They abstract away the complexities of accessing and modifying specific parts of an object by providing a uniform interface. They also enable easy composition of operations on different parts of an object. # Lens and Dynamical Systems According to [[David Jaz Myers]], Lens helps model dynamical systems by providing a way to represent and manipulate time-dependent data. Lens is a functional programming library that allows for the manipulation of immutable data structures, which is crucial for modeling dynamical systems. Lens provides a set of abstractions, such as lenses, prisms, and traversals, that allow us to focus on specific parts of our data and perform transformations or computations on them. This is particularly useful when dealing with time-dependent data because it allows us to easily track changes over time. By using lenses, we can create a view into our data that only shows relevant information at a specific point in time. We can then use this view to perform computations or extract information about the system's behavior over time. Lens also provides mechanisms for composing lenses together and applying transformations on them. This allows us to build complex models of dynamical systems by combining multiple lenses and manipulating the underlying data. Overall, Lens helps model dynamical systems by providing a powerful framework for representing and manipulating time-dependent data. It enables us to focus on specific aspects of the system's behavior over time and perform computations or extract information from it. # Conclusion Overall, lenses provide an elegant framework for manipulating objects within categories by focusing on specific components while maintaining the integrity and structure of the original object. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Lens") ```