What is Irreductibility in Category Theory?

In category theory, a functor is called irreductible if it cannot be decomposed as a product of simpler functors. In other words, a functor is irreductible if it cannot be expressed as a composition of "simpler" functors, where simplicity is measured in terms of the number of morphisms that are involved in the composition.

For example, consider the category of sets, where the only morphisms are functions between sets. The identity functor, which simply returns the set unchanged, is an irreductible functor because it cannot be decomposed as a product of simpler functors. On the other hand, the functor that maps each set to its powerset is not irreductible because it can be decomposed as a product of simpler functors: the functor that maps each set to its underlying set, and the functor that maps each set to its powerset.

Ireductibility is an important concept in category theory because it is closely related to the notion of "primitive" objects or "basic" objects. In any category, there are certain objects that cannot be decomposed into simpler objects, and these objects are often referred to as primitive or basic. Similarly, there are certain functors that cannot be decomposed into simpler functors, and these functors are often referred to as irreductible.

In summary, irreductibility is a concept in category theory that refers to the idea that some functors cannot be decomposed into simpler functors. It is closely related to the notion of primitive or basic objects, and it is an important concept for understanding the structure of categories.