0morphisms

0morphisms, also known as zero morphisms, are a concept in category theory, a branch of mathematics that studies abstract structures and their relationships. A 0morphism is a morphism (a generalization of a function or mapping) that maps every element of the domain to a single, fixed element in the codomain. This fixed element is often referred to as the "zero" element, hence the name "0morphism."

In the context of additive categories, such as the category of abelian groups or the category of modules over a ring, 0morphisms play a significant role. They are the morphisms that correspond to the zero element in the hom-set (the set of morphisms between two objects). In these categories, the zero morphism from object A to object B is the unique morphism that sends every element of A to the zero element of B.

0morphisms are also used in the study of monoidal categories, where they serve as the unit for the tensor product. In a monoidal category, the tensor product of an object with the zero object (if it exists) is isomorphic to the zero object, and the zero morphism serves as the identity morphism for the tensor product.

In summary, 0morphisms are a fundamental concept in category theory, representing morphisms that map every element of the domain to a fixed element in the codomain. They have important applications in additive categories and monoidal categories.