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I haven't studied much category theory (my experience with it is essentially limited to browsing a few snippets off of Wikipedia), but I've read in the past about the basic notions of morphisms, functors, etc.

I've also read that, within theoretical computer science, type theory is often expressed in the language of category theory. One thing led to another, and this got me wondering: Can a type-cast in a language like C++ be considered akin to a functor between categories? For example, if I have a class D which inherits from a base class B, would the act of casting a D-pointer into a B-pointer be akin to invoking a forgetful functor?

I know that this is probably a pretty basic question (and probably in many ways trivial), but I always enjoy looking at different concepts from as many perspectives as possible. Thanks in advance!

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No. Types are not categories, they are objects in a category. Type casting therefore is just a morphism between two objects, much like any other function. However, the type constructor "pointer to (-)" is a functor, since any function $X \to Y$ can be lifted to give a function $\textrm{ptr}(X) \to \textrm{ptr}(Y)$ in a way that respects composition. Similarly, the type constructor "list of (-)" is a functor, and even better, a monad. – Zhen Lin Feb 26 '12 at 19:52
Thanks for the explanation! If you put the comment as an answer, I'll be happy to "accept" it. – Dan M. Katz Feb 26 '12 at 20:34
up vote 2 down vote accepted

Here is a very brief sketch of how type theory can be studied from the point of view of category theory.

The main idea is that the type system of a programming language is a category: to be more precise,

  • types are objects, and
  • functions of one variable are morphisms.

It is then clear that type casting, like any other function, is just a morphism in this category.

Still, this interpretation is somewhat unsatisfactory since we frequently deal with functions of many variables. There are two ways around this: we could either define the cartesian product of types, or we could define function types (i.e. currying). Or we could do both. This leads to the notion of a cartesian closed category. The semantics of most progamming languages are not easy to analyse by category-theoretic means, so we should focus on pure functional programming languages with no side effects.

Now, a functor $F : \mathcal{C} \to \mathcal{D}$ comprises the following data:

  • a mapping of objects of $\mathcal{C}$ to objects of $\mathcal{D}$, and
  • a mapping of morphisms $X \to Y$ in $\mathcal{C}$ to morphisms $F X \to F Y$ in $\mathcal{D}$, such that identities and compositions are preserved.

If $\mathcal{C}$ is a type system for a programming language, then a functor $\mathcal{C} \to \mathcal{C}$ can (sometimes) be regarded as a type constructor. Here are some examples of functors:

  • The trivial functor $\bot : \mathcal{C} \to \mathcal{C}$ that sends every type to the bottom type $\bot$ and every morphism to the identity map $\textrm{id} : \bot \to \bot$ is a functor. More generally one can construct a constant functor for any type in this way.

  • The cartesian product functor $- \times -: \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ takes two types to their cartesian product. This can be regarded as the type constructor for pairs.

  • The list functor $\textrm{List}(-) : \mathcal{C} \to \mathcal{C}$ takes a type $X$ to the type of lists of $X$. This is a functor: its action on morphisms is better known as higher-order function ‘map’.

The last one is an example of a monad. To explain what a monad is, we first need to know what a natural transformation is: given two functors $F, G : \mathcal{C} \to \mathcal{D}$, a natural transformation $\alpha : F \Rightarrow G$ is a family of morphisms $\alpha_X : F X \to G X$ indexed by objects $X$ of $\mathcal{C}$ such that for all morphisms $f : X \to Y$ in $\mathcal{C}$, we have $$\alpha_Y \circ F f = G f \circ \alpha_X$$ (This is usually expressed by means of a commutative diagram.)

For example, there is a natural transformation from the identity functor to the trivial functor $\bot$, because every type $X$ admits a unique function $X \to \bot$, and the uniqueness means that the naturality condition is automatically satisfied.

A monad is a functor $T : \mathcal{C} \to \mathcal{C}$ together with natural transformations $\eta : \textrm{id}_\mathcal{C} \Rightarrow T$ and $\mu : T^2 \Rightarrow T$ such that, for all types $X$, $$\begin{align} \mu_X \circ T \eta_X & = \textrm{id}_X & \mu_X \circ \eta_{T X} & = \textrm{id}_X & \mu_X \circ T \mu_X & = \mu_X \circ \mu_{T X} \end{align}$$

We think of monads as constructors for wrapper types via the natural transformation $\eta$. Monads have the property that a multiply-wrapped type can be reduced to a singly-wrapped type in a coherent way, via the natural transformation $\mu$. Concretely, for the list monad, we have:

  • $\eta_X : X \to \textrm{List}(X)$ is the function that takes an $x$ and returns the singleton list $\langle x \rangle$.
  • $\mu_X : \textrm{List}(\textrm{List}(X)) \to \textrm{List}(X)$ is the function that concatenates lists; so for example, $\mu_\mathbb{N} (\langle \langle 1, 2 \rangle, \langle 3, 4, 5 \rangle \rangle) = \langle 1, 2, 3, 4, 5 \rangle$.

It is easy to check that these satisfy the axioms for a monad.

Finally, we have the notion of an algebra for a monad. If $(T, \eta, \mu)$ is a monad, then a $T$-algebra is an object $X$ together with a morphism $\alpha : T X \to X$ such that $$\begin{align} \alpha \circ \eta_X & = \textrm{id}_X & \alpha \circ \mu_X & = \alpha \circ T \alpha \end{align}$$

We think of these as being very well-behaved fold functions. For example, the function $\Sigma : \textrm{List}(\mathbb{N}) \to \mathbb{N}$ which sums up a list of numbers makes $\mathbb{N}$ into an algebra for the list monad. One imagines that this associativity property can be used to automatically parallelise certain computations. On the other hand, the averaging function $\textrm{List}(\mathbb{R}) \to \mathbb{R}$ is not an algebra for the list monad because the result depends on how the list is partitioned into sublists.

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