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Concept
Initial algebra
Summary
In mathematics, an initial algebra is an initial object in the of F-algebras for a given endofunctor F. This initiality provides a general framework for induction and recursion.
Consider the endofunctor F : Set → Set sending X to 1 + X, where 1 is the one-point (singleton) set, the terminal object in the category. An algebra for this endofunctor is a set X (called the carrier of the algebra) together with a function f : (1 + X) → X. Defining such a function amounts to defining a point and a function X → X.
Define
and
Then the set N of natural numbers together with the function [zero,succ]: 1 + N → N is an initial F-algebra. The initiality (the universal property for this case) is not hard to establish; the unique homomorphism to an arbitrary F-algebra (A, [e, f]), for e: 1 → A an element of A and f: A → A a function on A, is the function sending the natural number n to fn(e), that is, f(f(...(f(e))...)), the n-fold application of f to e.
The set of natural numbers is the carrier of an initial algebra for this functor: the point is zero and the function is the successor function.
For a second example, consider the endofunctor 1 + N × (−) on the category of sets, where N is the set of natural numbers. An algebra for this endofunctor is a set X together with a function 1 + N × X → X. To define such a function, we need a point and a function N × X → X. The set of finite lists of natural numbers is an initial algebra for this functor. The point is the empty list, and the function is cons, taking a number and a finite list, and returning a new finite list with the number at the head.
In categories with binary coproducts, the definitions just given are equivalent to the usual definitions of a natural number object and a list object, respectively.
Dually, a final coalgebra is a terminal object in the category of F-coalgebras. The finality provides a general framework for coinduction and corecursion.
For example, using the same functor 1 + (−) as before, a coalgebra is defined as a set X together with a function f : X → (1 + X).
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