Category Theory definition

Set theory is about membership while category theory is about structure-preserving transformations

Category theory is a general theory of mathematical structures and their relations

A category is formed by two sorts of things,

• the objects of the category, and
• the morphisms, which relate two objects (the source and the target of the morphism).

One often says that a morphism is an arrow that maps its source to its target. This is a bit like tracing a path through a graph.

Morphisms can be composed if the target of the first morphism equals the source of the second one, and morphism composition has similar properties as function composition (associativity and existence of identity morphisms).

Morphisms are often some sort of function, but this is not always the case.

The second fundamental concept of category is the concept of a functor, which plays the role of a morphism between two categories C1 and C2:}

it maps objects of C1 to objects of C2 and morphisms of C1 to morphisms of C2 in such a way that sources are mapped to sources and targets are mapped to targets

A third fundamental concept is a natural transformation that may be viewed as a morphism of functors.

In typical business rules the mappings are from a set of input values to a set of output values. The values are typically strings, integers, reals, dates, times and booleans. But these mappings can be generalized.

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