Maths - Category Theory - F-Algebra and F-Coalgebra

I assume 'F' stands for functor.

Signature of an Algebra

We can think of a binary operation as taking two numbers and returning another number. So
a = b * c
is a mapping
(b,c) -> a

Since a,b & c are all of the same type (such as real numbers) we can show the type signature as:

T -> T²

where

Note: we have reversed the direction of the functor, we will see later that for an algebra the direction of the operation and the functor are reversed and for a coalgebra they are not.

So in the case of numbers, T is the set of all numbers so T² is the cartesian product of these sets.

So far we have just looked at one operation in an algebra, lets now look a whole algebra, an example might be groups. A group has one binary operation (multiplication), one unary operation (inverse) and one identity element. The binary operation has a domain type of T², the unary operation has a domain type of T and the identity has a domain type of 1. We can therefore give the signature functor F of an algebra as a polynomial:

F : T -> T² + T + 1

where

So in the reverse direction we can get functors like these:

0
1 -> T
0 1 2 3 4
0 1 2 3 4
T -> T
0 1 2 3 4
(0,0) (0,1) (0,2) (0,3) (0,4)
(1,0) (1,1) (1,2) (1,3) (1,4)
(2,0) (2,1) (2,2) (2,3) (2,4)
(3,0) (3,1) (3,2) (3,3) (3,4)
(4,0) (4,1) (4,2) (4,3) (4,4)
T² -> T
0 1 2 3 4

F-Algebra & F-Coalgebra

The functor 'F' that we have defined above is an endofunctor to/from a category C which contains the type T and its powers to allow the following:

F-Algebra F-Coalgebra

An F-Algebra consists of a pair: (T, α) which is a type and a function α which is defined as:

α : F T -> T

An F-Coalgebra consists of a pair: (T, β) which is a type and a function β which is defined as:

β : T -> F T

f-algebra f-coalgebra

 


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