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Concept
F-algebra
Summary
In mathematics, specifically in , F-algebras generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor F, the signature.
F-algebras can also be used to represent data structures used in programming, such as lists and trees.
The main related concepts are initial F-algebras which may serve to encapsulate the induction principle, and the construction F-coalgebras.
If is a , and is an endofunctor of , then an -algebra is a tuple , where is an of and is a -morphism . The object is called the carrier of the algebra. When it is permissible from context, algebras are often referred to by their carrier only instead of the tuple.
A homomorphism from an -algebra to an -algebra is a -morphism such that , according to the following commutative diagram:
Equipped with these morphisms, -algebras constitute a category.
The dual construction are -coalgebras, which are objects together with a morphism .
Classically, a group is a set with a group law , with , satisfying three axioms: the existence of an identity element, the existence of an inverse for each element of the group, and associativity.
To put this in a categorical framework, first define the identity and inverse as functions (morphisms of the set ) by with , and with . Here denotes the set with one element , which allows one to identify elements with morphisms .
It is then possible to write the axioms of a group in terms of functions (note how the existential quantifier is absent):
Then this can be expressed with commutative diagrams:
Now use the (the disjoint union of sets) to glue the three morphisms in one: according to
Thus a group is a -algebra where is the functor . However the reverse is not necessarily true. Some -algebra where is the functor are not groups.
The above construction is used to define group objects over an arbitrary category with and a terminal object .
Official source
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