Fullness

Definition

Fullness is a property of functors in category theory, specifically relating to how well a functor captures the structure of morphisms between categories. A functor is said to be full if every morphism between objects in the target category can be represented as a morphism that comes from the source category, meaning it reflects all the relationships between objects faithfully.

5 Must Know Facts For Your Next Test

1. For a functor to be full, it must map every morphism from the source category to an existing morphism in the target category, capturing all possible relationships.
2. Fullness is often considered alongside other properties like faithfulness and completeness when evaluating functors.
3. A functor that is both full and faithful provides a strong correspondence between the structure of two categories, essentially making them equivalent in terms of morphisms.
4. An example of a full functor can be seen in representations of groups where each homomorphism corresponds directly to a morphism in another category.
5. When studying adjunctions, understanding fullness helps in analyzing how structures are preserved or altered during the mapping between categories.

Review Questions

• How does fullness relate to the concept of morphisms between categories?
  • Fullness connects directly to how morphisms from one category are represented in another. A full functor ensures that for every morphism in the target category, there exists a corresponding morphism in the source category. This means that the functor captures all structural relationships defined by those morphisms, allowing for a rich representation of interactions between objects.
• In what ways does fullness complement other properties like faithfulness when examining functors?
  • Fullness and faithfulness are two important properties used to evaluate functors. While fullness ensures that all morphisms are represented, faithfulness guarantees that each morphism in one category corresponds to a unique morphism in another. Together, they provide a comprehensive understanding of how well a functor preserves the relationships and structures between categories, indicating the degree of similarity or equivalence between them.
• Evaluate the significance of fullness in the context of categorical equivalence and how it contributes to understanding mathematical structures.
  • Fullness plays a crucial role in establishing categorical equivalence, which occurs when two categories can be shown to have an identical structure through their functors. By ensuring that every morphism is represented, fullness allows mathematicians to draw meaningful connections between seemingly disparate categories. This not only enhances our understanding of the relationships between different mathematical structures but also provides tools for translating problems and concepts across various areas of mathematics.

Related terms

Functor:

A functor is a mapping between categories that preserves the structure of objects and morphisms, effectively translating relationships from one category to another.

Natural Transformation: A natural transformation is a way of transforming one functor into another while maintaining the structure of the categories involved, acting as a bridge between different functors.

Faithfulness: Faithfulness is a related property of functors where every morphism between objects in one category corresponds to a unique morphism in another category, but it does not require that all morphisms be represented.