Problem Detail: If the definition of Initial Algebra is: “An object is initial if there exists a unique morphism from the object to every object in the category” Why do we need such object, and could any one give an example ?
Asked By : M.M
Answered By : Romuald
An initial algebra is an initial object in the category of $F$-algebras for a given endofunctor $F : mathcal{C} rightarrow mathcal{C}$. This construction is widely used to gives semantics to data-structures in (functional) programming languages. Intuitively, the functor $F$ captures the “shape” of the data-structure (e.g., $F(X) = 1 + A times X$, with $A$ a fixed set for instance). The underlying set $mu F$ of the initial algebra $langle mu F, alpha rangle$ intuitively captures the set of syntactic expressions you can build by induction on the shape functor (e.g., for the previous functor with $A = {a,b }$, $mu F = {(1), (a,1), (b,1), (a,a,1), (a,b,1) ldots }$ is the set of finite lists of $A$-elements). The initiality is the key property to construct inductive function (see catamorphism). Some excellent references on this topic:
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Question Source : http://cs.stackexchange.com/questions/8985
