Contents

Idea

In computer science, originally in database theory, a concept called lenses is used to formally capture situations where some structure is converted to a different form – a view – in such a way that changes made to the view can be reflected as updates to the original structure. The same construction has been devised on numerous occasions (Hedges).

There are, in general, two different approaches to lenses:

• Lawful lenses are an algebraic structure which axiomatize product projections. The laws govern the ways views and updates relate. These are generalized into Delta lenses, which are more flexible lawful lenses.

• Lawless lenses, and in particular bi-directional or polymorphic lenses, separate the two directions of view and update so that the laws no longer type-check. These lawless lenses are used to organize bi-directional flow of data, and are generalized into optics in the functional programming literature.

An alternate generalization of lawless lenses is put forward in Spivak 19 as the Grothendieck construction of the fiberwise dual of an indexed category. This notion of lawless lens has been adopted in the context of categorical systems theory because they represent a bidirectional stateful computation which describes the way some systems expose and update their internal state. For instance, see (Myers), (Spivak-Niu), or (Hedges 21).

In this sense, lawless lenses are applications of two mathematical frameworks which are interesting in their own right: fibrations and optics (thus Tambara modules).

Definition

Let $(\mathbf{C}, 1, \times)$ be a category with finite products.

Definition

Let $S,V$ be objects of $\mathbf C$. A (lawful) lens $L$, with source $S$ and view $V$, is a pair of arrows $\mathrm{get} : S \to V$ and $\mathrm{put} : V \times S \to S$, often taken to satisfy the following equations, or lens laws:

1. (PutGet) the get of a put is the projection: $\mathrm{get} \mathrm{put} = \pi_0$.
2. (GetPut) the put for a trivially updated state is trivial: $\mathrm{put} \langle \mathrm{get}, 1_S \rangle = 1_S$.
3. (PutPut) composing puts does not depend on the first view update: $\mathrm{put}(1_V \times \mathrm{put}) = \mathrm{put} \pi_{0,2}$.

Remark

The two morphisms comprising a lens can also be called ‘view’ and ‘update’. More generally, they are referred to as the ‘forward’ and ‘backward’ parts of the lens.

Remark

Sometimes a lens satisfying all three laws is said to be lawful. Sometimes it is said that a well-behaved lens satisfies (1) and (2) and a very well-behaved lens satisfies also (3).

The category of lenses

Lenses (regardless of their lawfulness) organize in a category $\mathrm{Lens}(\mathbf C)$ whose objects are the same as $\mathbf C$ and whose morphisms $X \to Y$ are lenses with states $X$ and views $Y$. The identity lens is given by $(1_X, \pi_1) :X \to X$. Composition of $(\mathrm{get}_1, \mathrm{put}_{1}):X \to Y$ and $(\mathrm{get}_2, \mathrm{put}_{2}):Y \to Z$ is given by:

$\mathrm{get}_{12} = \mathrm{get}_1 \circ \mathrm{get}_2$
$\mathrm{put}_{12} = \mathrm{put}_1 \circ \langle \mathrm{put}_2 \circ \langle 1_Z, \mathrm{get}_1\rangle, 1_X \rangle \circ \langle 1_Z, \Delta_X \rangle$

The $\mathrm{put}_{12}$ morphism is probably easier to describe using generalized elements:

$\mathrm{put}_{12} : (z,x) \mapsto \mathrm{put}_1(\mathrm{put}_2(z, \mathrm{get}_1(x)), x)$

Remark

Crucially, associativity of this composition relies on naturality of the diagonals, which is a given in cartesian categories but not in more general monoidal categories. Optics are a sweeping generalization of lenses which overcomes this obstacle.

Moreover, the cartesian product of $\mathbf C$ endows $\mathrm{Lens}(\mathbf{C})$ of a monoidal product.

Properties

Proposition

Let $\mathbf{C} = Set$. Then every lens $L = (S, V, \mathrm{get}, \mathrm{put})$ is equivalent a “constant complement” lens whose Get is a product projection $\pi_{1} : C \times V \to V$ and whose Put is the function $\pi_{0,2} : C \times V \times V \to C \times V$ for some set $C$.

Proposition

Let $\mathbf{C}$ be a cartesian closed category. Lenses in $\mathbf{C}$ are coalgebras for a comonad (the store comonad) the generated by the adjunction:

Generalizations

There are many generalizations of lenses which have been proposed, however they can be broadly classified into those which satisfy analogues of the lens laws, and those without any axioms or laws.

Lenses without laws

• One generalization considers the lenses from the previous section as monomorphic by contrast to polymorphic lens which go between pairs of types, $\lambda: (S, T) \to (A, B)$, consisting of a view function, $v_{\lambda}: S \to A$, and an update function $u_{\lambda}: S \times B \to T$. Without further conditions, these are known as bimorphic lenses. To impose conditions comparable to the lens laws above requires that the types be related.

These sorts of lenses are generalized by Spivak 19. For a quick explanation of how these sorts of generalized lenses are of use in systems theory, see Myers20; for a longer explanation, see Chapter 2 of Myers.

• An optic generalizes the way lenses ‘remember’ state from the forward part to the backward part, avoiding the necessity of a cartesian structure by swapping it with a sufficiently rich actegorical context.

Lenses with laws

• Delta lenses are a generalization which does satisfy laws. Here we have categories $S$ and $V$ called the source and view, together with a Get functor $g : S \to V$ and a function $\varphi \colon S_{0} \times_{V_{0}} V_{1} \to S_{1}$ which takes a pair $(s \in S, u : gs \to v \in V)$ to a morphism

$\varphi(s, u) : s \to p(s, u)$ in $S$ where $p(s, u) = cod(\varphi(a, u))$ is the Put function. The function $\varphi$ must also satisfy three lens laws. When $S$ and $V$ are codiscrete categories, delta lenses are equivalent to a lens in Set; see (Johnson-Rosebrugh 2016, Proposition 4).

• A morphism between directed containers is another kind of generalised lens satisfying laws called update-update lenses; see (Ahman-Uustalu 2017, Section 5). These are equivalent to cofunctors.